Mesoscale Dynamics By Yuh-Lang Lin [Lin, Y.-L., 2007: Mesoscale Dynamics. Cambridge University Press, 630pp.]
Chapter 5 Orographically forced flows
Many well-known weather phenomena are directly related to flow over orography, such as 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, and orographically influenced storm tracks. A majority of these phenomena are mesocale and are induced by stably stratified flow over orography. Thus, understanding the dynamics associated with stably stratified flow over a mesoscale mountain is essential in improving the prediction of the above mentioned phenomena. In addition, understanding the dynamics of orographically forced flow will also help on different aspects of meteorology, such as turbulence which affect aviation safety, wind-damage risk assessment, pollution dispersion in complex terrain, and subgrid-scale parameterization of mountain wave drag in general circulation models.
5.1 Flows over two-dimensional sinusoidal mountains
Some fundamental properties of flow responses to orographic forcing can be understood by considering a two-dimensional, steady-state, adiabatic, inviscid, nonrotating, Boussinesq fluid flow over a small-amplitude mountain. The governing linear equations can be simplified from (2.2.14)-(2.2.18) to be
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u' 1 p' U U zw' 0 , (5.1.1) x o x
w' ' 1 p' U g 0 , (5.1.2) x o o z
u' w' 0, (5.1.3) x z
' N 2 U o w' 0. (5.1.4) x g
The above set of equations can be further reduced to Scorer’s equation (1954),
2w' l 2 (z)w' 0 , (5.1.5) where 2 2 / x2 2 / z2 is the two-dimensional Laplacian operator, and l is the
Boussinesq form of the Scorer parameter (Scorer 1949), which is defined as:
N 2 U l 2 (z) zz . (5.1.6) U 2 U
Equation (5.1.5) serves as a central tool for numerous theoretical studies of small- amplitude, two-dimensional mountain waves, which may also be interpreted as a vorticity
' ' equation upon being multiplied by U (Smith 1979). 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', represents the rate of vorticity production by the redistribution of the background vorticity (U z ). In the extreme case of very small Scorer parameter, (5.1.5) reduces to irrotational or potential flow,
2w' 0 . (5.1.7)
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As discussed in Chapter 3 [(3.5.22)], the buoyancy force is negligible in this extreme case. 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.
For this particular case, there is no drag produced on the mountain since the fluid is inviscid.
In order to simplify the mathematics of the steady state mountain wave problem, one may assume that U (z) and N (z) are independent of height, and a sinusoidal terrain
h(x) hm sinkx , (5.1.8) where hm is the mountain height and 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) . (5.1.9) u U u' dx
For a small-amplitude mountain, this leads to the linear lower boundary condition
dh w' U at z 0, (5.1.10) dx or
w'(x,0) U hmk coskx at , (5.1.11) for flow over a sinusoidal mountain as described by (5.1.8). Due to the sinusoidal nature of the forcing, it is natural to look for solutions in terms of sinusoidal functions,
w'(x,z) w1(z) coskx w2(z) sin kx . (5.1.12)
Substituting the above solution into (5.1.5) with a constant Scorer parameter leads to
2 2 wizz (l k )wi 0 , i 1, 2 . (5.1.13)
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As discussed in Chapter 3 [(3.5.7)], two cases are possible: (a) l2 k 2 and (b) l2 k 2
. The first case requires N /U k or Na/2 U , 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 ) /(2 / 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/ 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, (5.1.13) can be rewritten as
2 2 wizz (k l )wi 0 , i 1, 2 . (5.1.14)
The solutions of the above second-order differential equation with constant coefficient may be obtained
z z wi Aie Bie , , (5.1.15) where
k2 l2 . (5.1.16)
Similar to that described in Section 3.4, the upper boundedness condition requires Ai 0 because the energy source is located at z 0. Applying the lower boundary condition,
(5.1.11), and the upper boundary condition ( ) to (5.1.15) yields
B1 U hmk; B2 0 . (5.1.17)
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This gives the solution,
k 2 l 2 z w'(x, z) w1(z)coskx Uhmke coskx , (5.1.18)
The vertical displacement ( ) is defined as w'/ D Dt which reduces to
D w' U (5.1.19) Dt x for a steady-state flow.
Equation (5.1.18) can then be expressed in terms of ,
1 x 2 2 w'dx h sinkx e k l z . (5.1.20) U 0 m
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 as discussed in Section 3.5. 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 (5.1.18),
2 2 k 2 l 2 z u' U hm k l sin kx e , (5.1.21)
2 2 2 k 2 l 2 z p' oU hm k l sin kx e , (5.1.22)
2 k 2 l 2 z ' (oN / g) hm sin kx e . (5.1.23)
The maxima and minima of u', p ', and ' 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, (5.1.1) with U z 0 , or (5.1.22), a low
(high) pressure is produced over the mountain peak (valley) where maximum (minimum)
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wind is produced. Note that (5.1.1) 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). The mountain drag can be computed either from the horizontal pressure force on the mountain over a wavelength,
k / k dh D p'( x , z 0) dx , (5.1.24) 2 / k dx or equivalently, as the negative of the vertical flux of horizontal momentum (momentum flux) in the wave motion,
k / k D o u'w'dx . (5.1.25) 2 / k
Note that the Eliassen and Palm’s theorem, (4.4.10), indicates that the vertical flux of horizontal momentum in a steady-state flow is negatively proportional to the vertical energy flux, p'w' (where the overbar denotes the average over a wavelength).
In the second case, l 2 k 2 , the flow response is completely different. This case requires N /U k or Na/2 U . As discussed in Section 3.5, 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-1and
N = 0.01 s-1, the terrain wavelength should be larger than 6.3 km. Since
(a /U ) /(2 / 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, (5.1.13) can be written as
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2 2 2 2 wizz m wi 0 , m l k , i 1, 2 . (5.1.26)
We look for solutions in the form
wi (z) Ai sinmz Bi cosmz, i 1, 2 . (5.1.27)
Substituting (5.1.27) into (5.1.12) leads to
w'(x, z) C cos(kx mz) Dsin(kx mz) E cos(kx mz) F sin(kx mz) . (5.1.28)
In the above equation, 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
have a positive vertical energy flux and should be retained since the energy source in this case is located at the mountain surface. This satisfies the Sommerfeld radiation boundary condition, as discussed in Section 4.4. Thus, the solution requires
E F 0 . This flow regime is characterized as the upward propagating wave regime, as discussed in Chapter 3. As in the first case, the lower boundary condition requires
C Uhmk, D 0 . (5.1.29)
This leads to
w'(x, z) Uhmk cos(kx mz ) . (5.1.30)
Other variables can be obtained through definitions or the governing equations,
(x, z) hm sin(kx mz) , (5.1.31)
u'(x, z) Uhmmcos(kx mz ) , (5.1.32)
2 p'(x, z) oU hmmcos(kx mz) , and (5.1.33)
N 2 h '(x, z) o m sin(kx mz) . (5.1.34) g
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The vertical displacement of the flow, and the maxima and minina of u' , p' , and ' are depicted in Fig. 5.1b. 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 (5.1.32) or the
Bernoulli equation (5.1.1), 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. The mountain drag can be calculated either from (5.1.24) or (5.1.25) to be
1 D U2 h 2 k l 2 k 2 . (5.1.35) 2 om
The positive wave drag on the mountain is produced by the high pressure on the windward slope and the low pressure on the lee slope. This also can be understood through (5.1.25) and the out-of-phase relationship of u' and w' over the windward and lee slopes, as shown in Fig. 5.1b.
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 . (5.1.36)
The flow pattern repeats itself in the vertical with a wavelength of z 2 /l 2U / 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
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by letting l k , which leads to a 2U / N . The relation among the mountain waves discussed in this subsection is sketched in Fig. 5.2.
5.2 Flows over two-dimensional isolated mountains
5.2.1 Uniform basic flow
The mountain wave problem in Section 5.1 may be extended to be more realistic by assuming an isolated mountain. Taking the one-sided Fourier transform (Appendix 5.1) of (5.1.5) yields
2 2 wˆ zz (l k )wˆ 0. (5.2.1)
The Fourier transform of the linear lower boundary condition, (5.1.10), is
wˆ(k, z 0) ikU hˆ(k) . (5.2.2)
For constant Scorer parameter, the solution of (5.2.1) can be written into two parts,
2 2 wˆ(k, z) wˆ(k,0) ei l k z for l 2 k 2 and (5.2.3a)
2 2 wˆ(k, z) wˆ(k,0) e k l z for l 2 k 2 . (5.2.3b)
Taking the inverse one-sided Fourier transform (Appendix 5.1) of (5.2.3) yields the solution in the physical space,
l 2 2 2 2 wxz'(,)2Re ikUhke ˆˆ() i l k z edk ikx ikUhke () k l z edk ikx , (5.2.4) 0 l where Re represents the real part. The first integration on the right hand side of (5.2.4) represents the upward propagating wave which satisfies the upper radiation boundary condition, while the second integration represents the evanescent wave which satisfies the boundedness upper boundary condition. Note that (5.2.4) is for a continuous spectrum of
Fourier modes, instead of just one single mode as considered in Section 5.1.
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For simplicity, let us consider a bell-shaped mountain or the Witch of Agnesi mountain profile,
h a2 h(x) m , (5.2.5) x2 a2 where hm is the mountain height and a is the half-width where the mountain height is hm / 2 . The advantage of using a bell-shaped mountain lies in that its one-sided Fourier transform (Appendix 5.1) is in a simple form,
h a hˆ(k) m eka, for k 0 . (5.2.6) 2
The Fourier transform for any k is (hma / 2)exp( k a) . First, we will center our discussion on the extreme case with l 2 k 2 (i.e., al 1 or Na U ). Note that for bell-shaped mountains, we assume ka1/ , instead of k = 2/a for sinusoidal mountains.
As discussed earlier, the flow becomes a potential flow in which the buoyancy plays a negligible role. In this case, (5.2.4) can be approximated by
ˆ kz ikx hma ka kz ikx w'(x, z) 2Re U ik h(k) e e dk 2Re U ik e e e dk . 0 0 2
(5.2.7)
Since w U / x , the Fourier transform of can be obtained from that of wˆ ,
wˆ(k, z) ˆ(k, z) . (5.2.8) ikU
Substituting (5.2.7) into (5.2.8) leads to
k() z a ix hm a() z a (x , z ) hm a Re e dk . (5.2.9) 0 x22() z a
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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 in Fig. 5.3a.
Second, let us consider another extreme case: l 2 k 2 (i.e., al 1 or Na U ).
As discussed in the previous section, the vertical acceleration due to the buoyancy force plays a dominant role. In this case, the solution (5.2.4) can be approximated by
ˆ ilz ikx hma ka ilz ikx w'(x, z) 2Re U ikh(k)e e dk 2ReU ik e e e dk . (5.2.10) 0 0 2
Similarly, the vertical displacement can be obtained,
h a ka i(kx lz) h a(a coslz xsin lz) (x, z) 2Re m e e dk m . (5.2.11) 0 2 x2 a2
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 2U / N . Without the Boussinesq approximation, the above solution becomes
1/ 2 h a(acoslz xsin lz) (x, z) s m , (5.2.12) 2 2 (z) x a where s is the air density near surface. Equation (5.2.12) indicates that the wave amplitude will increase with a decreased air density of the basic flow. That is, the wave amplitude will increase at higher altitudes since air density decreases with height in a stably stratified flow. This helps explain the wave amplification in the higher atmosphere, such as large-amplitude gravity waves in the stratosphere. As described in the previous section, other fields can be obtained by the governing equations, (5.1.1)-
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(5.1.4) with U z 0 . The wave drag on the mountain surface in this hydrostatic limit can be obtained by applying the Parseval theorem (Appendix 5.1),
ˆ* dh dh 2 D p'(x, z 0) dx pˆ (k,0) dk oUNhm , (5.2.13) dx dk 4 where h* is the complex conjugate of h. The momentum is transferred to a level where the wave breaks down, which is not included in the linear theory.
Third, an asymptotic solution can be obtained for the case with lk22 (i.e., al 1 or
Na U ). In this case, all terms of the vertical momentum equation, (5.1.2) are equally important. Both asymptotic methods and numerical methods have been applied to solve the problem. In the following, we apply the stationary phase method to this particular problem. We look for solutions far downstream, x in (5.2.4). In this limit, the second term on the right side of (5.2.4) approaches 0 due to fast oscillation of exp(ikx) , according to the Riemann-Lebesgue Lemma (Appendix 5.1). For large x , we have
l (x, z) 2Re hˆ(k)ei(k)dk , (5.2.14) 0 where
(k) l 2 k 2 z kx (5.2.15) is a phase function. Based on the stationary phase method, we will look for a particular k* such that
d 0 at kk* , (5.2.16) dk
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where k * is called the point of stationary phase. With large x or z , exp(i) will oscillate rapidly and, therefore, will approach 0, according to Riemann-Lebesgue Lemma.
However, near , the contribution to the integration by exp(i ) still remains because is approximately constant. Substituting the phase function (5.2.15) into (5.2.16) leads to the influence function,
z l2 k *2 , (5.2.17) xk* in the region near . Taking the Taylor’s series expansion of (k) near gives
2 *21 (k ) ( k ) k k ... , (5.2.18) 2 kkk* 2! where k k k* . The second term on the right side of the above equation disappears due to the definition of in (5.2.16). Thus, (5.2.14) becomes
* l 2 (x , z ) 2 Re hˆ ( k*() ) eik eikkk /2 dk . (5.2.19) 0
For a bell-shaped mountain,
2 *2 3/ 4 ka*()lk 2 *2 * (x , z ) 2 hm ae1/ 2 cos l k z k x , (5.2.20) lz 4 where
l k * . (5.2.21) (zx / )2 1
Figure 5.4 shows an example of a flow over a ridge of intermediate width ( lk22 ) where the buoyancy force is important, but not so dominant that the flow becomes hydrostatic. The nearly periodic waves located to the upper right of the mountain are the dispersive tail of nonhydrostatic waves with k less than, but not much less than l .
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In fact, the influence function, (5.2.17), is related to the energy propagation associated with the mountain waves. The group velocity (cgm) in the frame of reference fixed with the mountain can be obtained from (3.5.11),
cgm U i k, (5.2.22) km where m stands for mountain and
Nk . (5.2.23) km22
Substituting (5.2.23) into (5.2.22) leads to
Nm2 Nkm cgmUU i c ga 2 2 3/ 2 i 2 2 3/ 2 k, (5.2.24) ()()k m k m
where cga is the group velocity relative to the air. Furthermore, the requirement of stationary waves, cpx U 0 , implies
N U . (5.2.25) km22
In (5.2.23), the negative sign is chosen in order to obtain positive cgz by assuming positive k and m due to the use of one-sided Fourier transform. The relationship among
cpxi , cgm and cga is sketched in Fig. 5.5. The upstream phase speed of the mountain wave is exactly equal to and opposite of the basic wind speed. The wave energy propagates upward and upstream relative to the air, but is advected downstream by the basic wind. Thus, relative to the mountain, the energy associated with the mountain waves propagates upward and downstream. The slope of the group velocity can be obtained by substituting U of (5.2.25) into (5.2.24) and then calculating the slope,
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c m N 2 l2 k 2 z gz . (5.2.26) 2 2 1 cgx k U k k x
In deriving the second equality, we have used (5.2.25), while in deriving the last equality, we have used (5.2.17) near the point of stationary phase, i.e. kk * . Therefore, the point of stationary phase is the value of k corresponding to a wave with a group velocity beam as shown in Fig. 5.5. Waves are found downstream since the horizontal group velocity is less than the phase speed.
For general cases, such as l 2 k 2 or l 2 k 2 , it is not easy to obtain analytical solutions from (5.2.4). With the advancement of numerical techniques, such as the Fast
Fourier Transform (FFT) and computers, solutions can be approximately obtained numerically with the implementation of proper boundary conditions.
5.2.2 Basic flow with variable Scorer parameter
In the real atmosphere, the basic wind and stratification normally vary with height.
To study the mountain waves produced by this type of basic flow, we assume that the
Scorer parameter, (5.1.6), is a slowly-varying function of z . In this situation, we expect to find a solution of (5.2.1) in form of,
wˆ ( k , z ) A ( k , z ) ei(,) k z , (5.2.27) where A(,)kz is a slowly varying amplitude function, and (k, z) is the slow-varying phase function. Substituting (5.2.27) into (5.2.1) yields
2 2 2 AAAAAz (l k ) i ( zz 2 z z ) zz 0. (5.2.28)
The last term makes a minor contribution and can be neglected, since A(,) k z is a slow- varying function of z. Thus, the above equation reduces to
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22 z lk, and (5.2.29)
(A 2 ) 0 . (5.2.30) z z
Combining the above two equations leads to
A 2l 2 k 2 constant . (5.2.31)
For long (hydrostatic) waves (l 2 k 2 ), the above equation reduces to
A2l constant . (5.2.32)
This implies that the amplitude of the vertical velocity increases (decreases) significantly in regions of weak (strong) stratification or strong (weak) wind. For example, the mountain wave tends to steepen when it propagates to the region below a jet stream or a jet streak since the basic wind speed increases there. Note that in applying (5.2.27) to solve the problem, and in neglecting the last term of (5.2.28), we have implicitly adopted a first-order WKBJ approximation. A second-order WKBJ approximation has been used to calculate wind profile effects on mountain wave drag (e.g., Teixeira and Miranda
2006). It is necessary to extend the WKBJ approximation to second order for these effects to be taken into account.
Based on (5.2.32) and previous discussions, waves may amplify in certain layers due to: (a) weaker stratification, (b) stronger wind, such as a jet stream or jet streak, (c) nonlinear steepening, and (d) abrupt decrease in the mean density, leading to an increase of s / (z) , in (5.2.12).
5.2.3 Trapped lee waves
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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, such as those shown in
Fig. 5.6. This type of wave differs from the dispersive tails in Fig. 5.4 in that it is located in the lower atmosphere and there is no vertical phase tilt. It will be shown below that this type of trapped lee waves, or resonance waves, occurs when the Scorer parameter decreases rapidly with height (Scorer 1949).
The dynamics of trapped lee waves may be understood by considering a two-layer stratified fluid system. The wave equations for the vertical displacement in Fourier space may be written in a form similar to (5.2.1),
2 2 ˆzz [l1 k ] ˆ 0 for H z 0 and (5.2.33a)
2 2 ˆzz [k l2 ] ˆ 0 for 0 z . (5.2.33b)
2 2 2 In this two-layer fluid system, we have assumed that l2 k l1 . For convenience, the ground and the interface of the lower and upper layers are assumed to be located at z H and z 0, respectively. The free wave solutions may be written as
ˆ 1(k , z ) C cos z sin z and (5.2.34a)
z ˆ2 (k , z ) C e , (5.2.34b)
2 2 2 2 where l1 k , k l2 and C is a constant coefficient to be determined by the lower boundary condition. The boundedness upper boundary condition has been applied to exclude the exp(z )term, and the kinematic and dynamic boundary conditions at the interface, i.e. the continuities of wˆ and wˆz at z 0 , have also been applied. Without enforcing a lower boundary condition, (5.2.34) represents free waves associated with this
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two-layer fluid system. The resonance waves are obtained by seeking the zeros of
(5.2.34a) with z H ,
cot H / . (5.2.35)
* The resonance wave number ( kr ) may be obtained by solving the above equation either numerically or graphically. The criterion for the existence of one or more resonance waves may be obtained (Scorer 1949):
2 2 2 l1 l2 . (5.2.36) 4H 2
A more general criterion for resonance waves of the nth mode is
22 (2nn 1) 22 (2 1) ()ll12 . (5.2.37) 22HH
The above criterion implies that in order to have resonance (lee) waves, the Scorer parameter in the lower layer must be much greater than that in the upper layer. In other words, the lower layer must be more stable or with a much slower basic wind speed than the upper layer.
In order to obtain a complete solution of the boundary value problem for a specific obstacle, we may apply the linear lower boundary condition,
ˆ ˆ1(,)()k H h k . (5.2.38)
Substituting the above equation into (5.2.34) and taking the inverse Fourier transform of
ˆ(k, z) leads to the forced wave solution in the lower layer,
ikx hˆ( k )(cos z ( / )sin z ) e 1(x , z ) 2 Re dk . (5.2.39) 0 (cosHH ( / )sin )
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The singularity in the denominator of the above equation corresponds to the resonance mode that will produce lee waves. Equation (5.2.39) can be solved asymptotically or numerically with a given mountain-shape function (Scorer 1949; Smith 1979). Figure
5.7 shows lee waves simulated by a nonlinear numerical model for a two-layer airflow over a bell-shaped mountain. 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, which will be further discussed in subsection 5.4.2 along with lee vortices.
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.
5.3 Nonlinear flows over two-dimensional mountains
As discussed in Sections 5.1 and 5.2, the response of a stably stratified flow over a two-dimensional mountain ridge has been studied extensively since the 1960’s. In
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particular, the linear dynamics are fundamentally understood, especially due to the development of linear theories in earlier times. 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,
F U/ Nh , 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. Note that the reciprocal of the Froude number, Nh/ U , has also been used as a control parameter and is known as nondimesnional mountain height. In the text, we will use these two parameters interchangeably.
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.
In this section, we will begin with the discussion of a nonlinear theory developed by
Long (1953), then discuss the two-dimensional flow regimes for a continuously stratified flow over a two-dimensional mountain with the help of nonlinear numerical models, and the generation mechanisms of severe downslope winds and wave breaking.
5.3.1 Nonlinear flow regimes
The governing equation for the finite-amplitude, steady state, two-dimensional, inviscid, continuously and stably stratified flow may be derived (Long 1953):
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2 2211de N ( ) 0 , (5.3.1) e dz z2 U 2 where (x, z) z zo is the streamline deflection at (x, z) from its far upstream, undisturbed height zo ; U and N are the far upstream basic flow speed and Brunt-
2 Vaisala frequency, respectively, at height zo , and eU (1/ 2)o is the kinetic energy of the upstream flow. In deriving (5.3.1), it has been assumed that there is no streamline deflection far upstream. In order to solve the above nonlinear equation, (5.3.1), we must specify e . Under the special situation de/0 dz and when the flow is Boussinesq, which assumes that is approximately constant and U (z) and N(z) are effectively constant, (5.3.1) becomes a linear Helmholtz equation,
2 l 2 0, (5.3.2) where l N /U is the Scorer parameter of the basic flow far upstream. The nonlinear lower boundary condition for (5.3.2) is given by
(x, z) h(x) at z h(x) , (5.3.3) where h(x) is the height of the mountain surface. In other words, 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, such as (5.1.10). Equation (5.3.2) with the lower boundary condition (5.3.3) forms Long’s model, in which the steady-state nonlinear flow is remarkably described by a linear differential equation with constant coefficients. In fact, (5.3.2) is exactly the same differential equation which applies to infinitesimal perturbations adopted in many linear theories and discussed earlier in this chapter. The appropriate upper boundary condition for a semi-infinite fluid, such as the
21
atmosphere, is the radiation or boundedness condition, similar to (5.2.3) in the Fourier space for a uniform basic flow over an infinitesimal mountain.
Following the procedure for treating linear flow over small-amplitude mountains, we make the Fourier transform of (5.3.2),
ˆ 2 2 ˆ zz (l k ) 0. (5.3.4)
The general solution for the above equation is
ˆˆ (ke ,0) imz for l k and (5.3.5a)
ˆˆ (ke ,0) z for l k , (5.3.5b) where m l22 k , and kl22. Note that the upper radiation and boundedness conditions have been applied to (5.3.5a) and (5.3.5b), respectively, while the linear lower boundary condition has been applied at z 0, instead of at z h(x) . The streamline deflection in the physical space can then be obtained by taking the inverse Fourier transform
l (,)Rexz ˆˆ (,0) keedkimz ikx (,0) keedk z ikx , (5.3.6) 0 l which may be obtained numerically, as with the Fast Fourier Transform numerical technique. Other dynamical variables may be derived,
2 N 2 g u ; w ; o 1 ; N , (5.3.7) z x gU o z where is the streamfunction defined as Uz() . The exact nonlinear lower boundary condition, (5.3.3), can be implemented using an iterative method (e.g., Laprise and Peltier
1989a).
22
Figure 5.8 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. As mentioned earlier, the nondimensional mountain height is a measure of the nonlinearity of the continuously stratified flow, which equals the reciprocal of the Froude number (
U / Nh ). 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 shear unstable (Laprise and Peltier 1989b). The vertical streamline marks the approximate limit of applicability of Long’s model. For the hydrostatic solution of Long’s model with a bell-shaped mountain subject to a nonlinear lower boundary condition, this critical value is Nh /U 0.85 (Miles and Huppert 1969).
Thus, for a continuously stratified, hydrostatic flow over a bell-shaped mountain, the flow may be classified as supercritical flow when U / Nh 1.18 ( Nh/ U 0.85) and as subcritical flow when U / Nh 1.18 ( Nh/ U 0.85). Note that in the literature it is often misquoted U/1 Nh as the regime boundary for supercritical and subcritical regimes for continuously stratified flow over mountains.
As discussed in Section 3.3, there exist five flow regimes in one-layer shallow-water system, based on the shallow water Froude number, F U/ gH , and the
nondimensional mountain height, M hm / H (Fig. 3.3). In a non-rotating, continuously stratified flow over a two-dimensional, bell-shaped mountain, three nondimensional control parameters may be identified: , h/ a , and Na /U . However, only two of them are independent. The parameter h/ a measures the steepness of mountain, and
Na /U is the nondimensional mountain width which measures the degree of hydrostatic
23
effect (the larger the more hydrostatic). In the hydrostatic limit ( Na /U ), the sole control parameter is the Froude number.
Figure 5.9a shows streamlines for Long’s solutions for flow over a bell-shaped mountain with a half-width (a) of 3 km with the nonlinear lower boundary condition
(5.3.3) applied. Internal waves tend to overturn in regions of reversed density gradient
(statically unstable), / z 0 , which corresponds to / z 1 from (5.3.7). The heights of critical steepening levels differ slightly from those predicted by linear theory for hydrostatic waves, zo (n 3/ 4)(2U / N) , where n is an integer, just over the crest of the topography (Laprise and Peltier 1989a). In Fig. 5.9a, the first steepening level for nonlinear, hydrostatic waves is about 2.36 km. With a narrower mountain, such as a = 1 km (Fig. 5.9b), a dispersive tail, caused by nonhydrostatic dispersion, is produced. The downstream displacement of the steepened region is caused by both the nonhydrostatic effect and the nonlinearity of the interior flow and the lower boundary condition. When
Na/U 1, the flow approaches the hydrostatic limit. This control parameter can be obtained by comparing the scales of 22wx'/ and N22 w'/ U of (5.1.5). Direct comparison of 2w'/ x2 and 2w'/ z 2 terms by scale analysis leads to the conclusion that h/a is a control parameter of nonhydrostatic effect. The Froude number, U/ Nh , can also be derived by comparing the scales of 22wz'/ and of (5.1.5).
Long’s nonlinear theory advances our understanding of orographically forced flow considerably. However, the constant upstream condition assumed by Long may not be necessarily consistent with the flow established naturally by transients, especially when blocking occurs (Garner, 1995). In the real atmosphere, turbulence will come into play and produce vertical mixing in a subcritical (overturning) flow. To simulate a subcritical
24
flow, one may consider using a laboratory tank experiment or adopting a nonlinear numerical model. As mentioned earlier, flow may become stagnant, where the total horizontal wind speed reduces to zero, in essentially two regions: in the interior of the fluid over the mountaintop or on the lee slope and along the upstream slope of the mountain.
Flow stagnation in a two-dimensional flow is responsible for flow recirculation, while stagnation in a three-dimensional flow is responsible for flow splitting. Flow stagnanation in the interior of the fluid is due to nonlinear wave steepening, which may lead to wave breaking and wave overturning over the lee slope, while the flow stagnanation at the upstream surface of the mountain is called flow blocking. Although the two-dimensional, nonrotating, hydrostatic flow may be simply classified as supercritical and subcritical regimes, as discussed above, the transient flow behavior becomes much more complicated. Figure 5.10 shows the time evolution for the and u' fields for a hydrostatic flow over a two-dimensional, bell-shaped mountain simulated by a numerical model at nondimensional times Ut / a 12.6 and 50.4 for F ranging from 0.5 to 1.3 . Four regimes are identified: (I) flow with neither wave breaking aloft nor upstream blocking (e.g., 1.12 F ), (II) flow with wave breaking aloft in the absence of upstream blocking (e.g., 0.9F 1.12 ), (III) flow with both wave breaking and upstream blocking, but where wave breaking occurs first (e.g., 0.6F 0.9), and (IV) flow with both wave breaking and upstream blocking, but where blocking occurs first
(e.g., 0.3F 0.6). Note that the exact Froude numbers separating these flow regimes might be different in other numerically simulated results because these numbers are
25
sensitive to some numerical factors, such as the grid resolution, domain size, numerical boundary conditions, and numerical scheme adopted in different numerical models.
In regime I (e.g., F = 1.3 in Fig. 5.10), neither wave breaking nor upstream blocking occurs, but an upstream propagating columnar disturbance does exist. The basic flow structure in regime I resembles linear mountain waves. Columnar disturbances are wave modes with constant phase in the vertical, which permanently alter the upstream temperature and horizontal velocity fields as they pass through the fluid (e.g.,
Pierrehumbert and Wyman 1985). A columnar disturbance may be generated by a sudden imposition of a disturbance, such as the impulsive introduction of a mountain in a uniform flow. Regime II (e.g., F = 1.1 in Fig. 5.10) resembles weakly nonlinear mountain waves. In this flow regime, an internal jump forms at the downstream edge of the wave-breaking region above the mountain, propagates downstream, and then becomes quasi-stationary. The region of wave breaking also extends downward toward the lee slope. After the internal jump travels farther downstream, a stationary mountain wave becomes established in the vicinity of the mountain above the dividing streamline, which is induced by wave breaking. A high-drag state is predicted in this flow regime. In addition, a vertically propagating hydrostatic gravity wave is generated by the propagating jump and travels with it. Along the lee slope, a strong downslope wind develops. Static and shear instabilities may occur locally in the region of wave breaking.
The computed critical Froude number for wave breaking is about 1.12, which agrees well with the value 1.18 found by Miles and Huppert (1969).
In regime III (e.g., F = 0.7 in Fig. 5.10), the internal jump over the lee slope propagates downstream in the early stage and then becomes quasi-stationary. Note that
26
the propagation of the downstream internal jump is sensitive to the upstream numerical boundary condition, which may cause the internal jump to retrogress upstream. To avoid this artificial effect from the numerical model, the upstream boundary should be placed far enough so as to effectively reduce its impact. Also, the layer depth of blocked fluid upstream is independent of the Froude number. In regime IV (e.g., F = 0.5 in Fig. 5.10), a significant portion of the upstream flow is blocked by the mountain. The presence of wave breaking aloft is not a necessary condition for upstream blocking to occur. A vertically propagating gravity wave is generated by the upstream reversed flow and travels with it. The speed of the upstream reversed flow is proportional to h/ a . The surface drag increases abruptly from regime I to II, while it decreases gradually from regime II (III) to III (IV).
Flow regimes may also be classified in different ways, depending upon particular characteristics. For example, two-dimensional, uniform flow over an isolated mountain has been classified as either a quasi-linear regime, high-drag state, or blocked state, based on Nh /U and NU / g (Stein 1992). In addition, the flow response of a three- dimensional flow over a long ridge is very different from that of a two-dimensional flow when is large. For example, the onset of wave breaking and the transition to the high-drag state in the three-dimensional flow was found to be accompanied by an abrupt increase in deflection of the low-level flow around the ridge (Epifanio and Durran 2001).
The increased flow deflection is produced at least in part by upstream-propagating columnar disturbances forced by the transition to the high-drag state.
5.3.2 Generation of severe downslope winds
27
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, bora over the Dinaric Alps, zonda over the Agentina mountains, berg wind in South Africa, Canterbury-nor’wester in New Zealand, halny wiatr in the mountains of
Poland, Santa Ana winds in southern California, and Diabolo winds in San Francisco Bay
Area. One well-known event is the 11 January 1972 windstorm which occurred in
Boulder, Colorado, and which reached a peak wind gust as high as 60 ms-1 and produced severe damage in the Boulder area (Fig. 3.4a).
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. These will be reviewed in the following.
a. 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, such as happens in Fig. 5.10 (F = 0.5, 0.7, and 1.1), in which severe downslope winds develop in a uniform flow over a bell-shaped mountain. The wave-breaking region is characterized by strong turbulent mixing (where Ri < 0.25), with a local wind reversal on top of it. As mentioned in Section 3.8, the 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. The lowest wave-induced critical level starts to develop at the height
28
z 3z / 4 2.36, 3.30, and 5.18 km for cases of F = 0.5, 0.7, and 1.1, respectively (at
Ut/ a 12.6 in Fig. 5.10), where z (= 2/UN) is the hydrostatic vertical wavelength. A supercritical flow with a severe downslope wind can be found over the lee slope under the wave breaking region, which undergoes a transition from subcritical flow over the upwind slope. The maximum perturbation wind over the lee slope is much higher than those predicted by linear and weakly nonlinear theories. At a later stage, the well-mixed layer (wake) deepens, the depth of internal hydraulic jump (critically steepened streamlines) extends to a great depth, the flow above the initial wave-induced critical level is less disturbed compared to that in the lower layer, and severe winds develop over the lee slope and below the well-mixed layer (Ut/ a 50.4 in Fig. 5.10).
The above example implies that 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.
This is shown by performing nonlinear numerical simulations for stratified flow over a bell-shaped mountain (Fig. 5.11). In these simulations, the basic flow reverses its direction at a prescribed critical level (zi). In the absence of shear instability associated with the basic flow, and when the basic-flow critical level is located at a nondimensional height of zi/z = 3/4 + n (n is an integer) 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. This leads to an extremely large Reynolds stress or surface drag and severe downslope winds (Figs. 5.11a and 5.11c). In other words, the flow is on resonance. On the other hand, when the basic flow critical level is located at a nondimensional height off zi/z = 3/4 + n, such as 1.15,
29
there is no wave resonance and no severe downslope winds generated (Fig. 5.11b).
Because the severe downslope winds are developed by resonance between upward and downward waves, this mechanism is referred to as the resonant amplification mechanism.
Based on numerical simulations with finer-grid resolutions, three distinct stages for the development of severe downslope winds are identified (Scinocca and Peltier 1993).
(1) Local static (buoyancy) instability develops when the wave steepens and overturns, thus producing a pool of well-mixed air aloft (Figs. 5.12a-b). (2) A well-defined large- amplitude stationary disturbance is generated over the lee slope. In time, small-scale secondary Kelvin-Helmholtz (K-H) (shear) instability develops in local regions of enhanced shear associated with flow perturbations caused by the large-amplitude disturbance (Figs. 5.12c-d). (3) The region of enhanced wind on the lee slope expands downstream, eliminating the perturbative structure associated with the large-amplitude stationary disturbance (Figs. 5.12e-f). The K-H instability dominates the flow in this mature windstorm state. Thus, static instability helps explain the initiation of wave- induced critical level and the downstream expansion of the severe downslope winds.
Once 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 in the lower uniform flow layer, similar to that in case 3 of Table 4.1 and Fig. 4.12b. Effects of the wave ducting on the development of high-drag states for a flow with uniform wind and constant static stability are illustrated in Fig. 5.13. Shortly after the occurrence of wave breaking, regions with local Ri < 0.25 form in the vicinity of the wave breaking (Fig.
5.13a). This turbulent mixing region is expanding downward and downstream due to strong nonlinear effects on the flow with low Richardson number near the critical level
30
(Fig. 5.14a). The turbulent mixing region expands downward by wave reflection, overreflection, and ducting from the wave-induced critical level and accelerates downstream by the nonlinear advevction (Fig. 5.13b).
Effects of wave reflection and/or overreflection are evidenced by the fact that the wave duct with severe downslope wind is located below the region of the turbulent mixing region. Note that the expansion of the turbulent mixing region provides a maintenance mechanism for the existence of the wave duct below it and above the lee slope, because the reflectivity in this region is about 1 according to linear theory.
Without this almost perfect reflector, the wave below cannot be maintained and would lose most of its energy due to dispersion. In fact, wave overreflection can occur, according to the wave ducting theory discussed in Chapter 4, through the extraction energy from the well-mixed region and thus contribute to the acceleration of downslope winds. In the absence of nonlinearity (Fig. 5.14b), the wavebreaking region does not expand downward to reduce the depth of the lower uniform wind layer. This, in turn, prohibits the formation of the severe downslope wind and internal hydraulic jump. These results indicate that the nonlinear wave ducting has contributed to the downward and downstream expansion of the turbulent mixing region.
b. Hydraulic theory
Based on the similarity of flow configurations of severe downslope windstorms and finite-depth, homogeneous flow over a mountain ridge, a nonlinear hydraulic theory was proposed to explain the development of severe downslope winds (Smith 1985). The hydraulic theory attributes the high-drag (severe-wind) state to the interaction between a
31
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 ( c in Fig. 5.15a). Assuming the upstream flow is uniform in U and N and the general flow is smooth, nondissipative, hydrostatic, Boussinesq and steady (Fig. 5.15a), the nonlinear, hydrostatic governing equation can be simplified from (5.3.1),
2 zz l 0 , (5.3.8)
The horizontal velocity can be derived from (5.3.7),
u U(1 z ) . (5.3.9)
The lower boundary condition is given by (5.3.3). By assuming no disturbance above
* the upper dividing streamline (Ho), the pressure at z Ho is constant, i.e. p(,) x Ho p .
If the air in the turbulent region is hydrostatic in the mean and well mixed with a density of c , the pressure along the lower branch of the dividing streamline is
* p(,) x Ho c p c g c , where c is the vertical displacement of the lower dividing
streamline ( H1 ). For a steady-state flow, the Bernoulli equation along zHoc can be written
2 p (1/ 2)u cgz constant. (5.3.10)
At z Ho c , we have
z 0 . (5.3.11)
By assuming a wave-like solution in the vertical,
(x, z) A(x) coslz B(x)sin lz , (5.3.12) the nonlinear solution for high-drag state can be obtained,
32
~ ~ ~ ~ ~ h c[cos(H o c h)], (5.3.13a)
~ ~ ~ ~ A c cos(H o c ) , and (5.3.13b)
~ ~ ~ ~ B c sin(H o c ) , (5.3.13c) where hx() is the terrain height function and all coefficients and parameters are nondimensionalized by l ( N /U ) and denoted by tilde’s (“ ~ ”). The above solution ~ ~ can be solved graphically or numerically as long as h and H o are known.
Figure 5.15 shows an example of a severe downslope windstorm simulated by a hydraulic theory with F 1.0 . The descent of the lower dividing streamline begins over the point where the mountain begins to rise and becomes more rapid over the mountain peak. The final downward displacement of the dividing streamline is a large fraction of the initial layer depth. The flow speed after transition to supercritical flow over the lee slope from subcritical flow over the upslope is greatest near the surface and is several times the upstream value. The flow shown in Fig. 5.15b is qualitatively similar to the
1972 Boulder windstorm observations (Fig. 3.4a). In addition to the above solution, the strength of the transitional flow can be measured by the pressure drag on the mountain per unit length,
N 2 D o (H H )3 . (5.3.14) 6 o 1
The hydraulic theory of severe downslope winds was confirmed by numerical experiments of stratified fluid flow (e.g., Durran and Klemp 1987; Bacmeister and
Pierrehumbert 1988) and laboratory tank experiments (e.g., Rottman and Smith 1989).
Note that in order to apply the hydraulic theory to the prediction of the steady-state flow over a mountain, it is necessary to specify the initial height of the dividing streamline
33
line. Thus, the dividing streamline height cannot be determined a priori if the critical level is induced by wave breaking. This, in turn, implies that the hydraulic model is limited to the consistent check of a severe wind state and cannot be used for prediction.
c. Applications of resonant amplification and hydraulic theories
Some discrepancies have been found between the resonant amplification and hydraulic theories of severe downslope windstorms. One discrepancy is the different critical level heights for high-drag (severe wind) states predicted by these two theories.
The resonant amplification theory predicts the wave-induced critical (wave breaking)
level at a height of zn/z 3/ 4 (n an integer), which helps produce severe downslope winds at later times. On the other hand, the hydraulic theory predicts ciritical level
heights falling within the range of zn/z 1/ 4 to 3/4 n during a high-drag state.
This discrepancy appears to be caused by different stages of the severe downslope wind state being used for prediction. In fact, in earlier stages of a high-drag state, the resonant amplification theory is consistent with weakly nonlinear theories which indicate that the initiation of a high-drag transitional flow begins with linear resonance (Grimshaw and
Smyth 1986), and with nonlinear numerical simulations which indicate that the lowest initial wave-induced critical level is near 3/4 (Lin and Wang 1996). It can also be seen clearly from Fig. 5.10 that the wave-induced critical level for a severe-wind state is shifted to a lower level at later time. Therefore, it appears that the resonant amplification theory focuses on the ealier stage of severe downslope wind development, while the hydraulic theory focuses on the later stage.
34
Part of the discrepancies may be related to the usage of critical level height as the control parameter to determine a high-drag state, as often adopted in many previous studies. Based on some numerical experiments, the lower uniform flow layer depth appears to be a more appropriate scale to use (Wang and Lin 1999). Figure 5.16 indicates that the high-drag state is sensitive to the lower stable layer depth (Durran 1986a). Using the lower layer depth as the control parameter, predictions of both high- and low-drag states from several previous numerical studies are shown to be consistent, and the high- drag state does depend on the mountain height, which is consistent with the hydraulic theory. In addition, some discrepancies among previous studies result from the choice of different Richardson numbers and basic flow velocity profiles (e.g., Teixeira et al. 2005).
5.4 Flows over three-dimensional mountains
Although the two-dimensional mountain wave theories discussed in previous sections 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. In this section, we will discuss a linear theory of a stratified flow past an isolated mountain, as well as the generation of lee vortices in a nonlinear flow over an isolated mountain.
5.4.1 Linear theory
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In the following, the two-dimensional, linear mountain wave theory developed in
Section 5.2.1 is extended to three-dimensional flow over an isolated mountain. Consider a steady state, small-amplitude, adiabatic, inviscid, nonrotating, stratified, Boussinesq fluid flow with uniform basic velocity (U) and Brunt-Vaisala frequency (N) over a three- dimensional topography h(x, y) . The governing linear equations can be derived from
(5.1.1)-(5.1.4),
u' 1 p' U 0, (5.4.1) x o x
v' 1 p' U 0, (5.4.2) x o y
w' ' 1 p' U g 0, (5.4.3) x o o z
u' v' w' 0 , (5.4.4) x y z
' N 2 U o w' 0. (5.4.5) x g
Using (5.1.19), the above equations can be combined into a single equation of ,
2 2 N 2 xx H 0 . (5.4.6) U 2
Equation (5.4.6) can be solved by taking the double Fourier transform in x and y to obtain
2 ˆzz m ˆ 0 , (5.4.7) where
m2 K 2( N 2 / k 2 U 2 1) , (5.4.8)
36
and K k 2 l 2 is the horizontal wavenumber. The double Fourier transform pair is defined as
1 ˆ(k,l, z) (x, y, z)ei(kxly)dxdy , and (5.4.9a) 4 2
(x, y, z) ˆ(k,l, z)ei(kxly)dkdl . (5.4.9b)
The solution to (5.4.7) in the Fourier space can be found
ˆ(k,l, z) ˆ(k,l,0)eim(k,l)z . (5.4.10)
Similar to the two-dimensional mountain wave theory, as discussed in section 5.2, there exist two flow regimes: (I) N2/1 k 2 U 2 , and (II) N2/1 k 2 U 2 . For upward propagating waves (regime I), the sign of m must be the same as the sign of k , in order to satisfy the upper radiation condition. On the other hand, for evanescent waves (regime II), the positive root of (5.4.8) must be chosen, i.e.
m (k,l)z ˆ(k,l, z) ˆ(k,l,0)e i , (5.4.11)
2 2 2 where mi is defined as K1/ N k U . The linear lower boundary condition is
(x, y, z 0) h(x, y) , (5.4.12) which can be transformed into the Fourier space,
ˆ(k,l,0) hˆ(k,l) . (5.4.13)
From the definition of inverse Fourier transform and (5.4.13), we have
(x, y, z) hˆ(k,l)eim(k,l)zei(kxly)dkdl . (5.4.14)
Now let us consider a three-dimensional (circular) bell-shaped mountain
h h(x, y) ; r x2 y2 , (5.4.15) (r 2 / a2 1)3/ 2
37
where h and a are the mountain height and horizontal scale, respectively. The Fourier transform of (5.4.15) is,
ha2 hˆ(,) k l eaK . (5.4.16) 2
The problem may be further simplified by using the hydrostatic approximation, i.e. neglecting the first term of (5.4.3). Note that under the hydrostatic approximation, we require that Na /U 1. The solution, (5.4.14), may be reduced to a single integration by converting it into cylindrical coordinates, and asymptotic solutions for the flow aloft and the flow near the ground may thus be obtained (Smith 1980). Substituting (5.4.16) into (5.4.14) and nondimensionalizing it according to
(,)(/,/),(,)(xy xaya z NzUN /, /),(,,)(, U klK kalaKa , ) , (5.4.17) yields
1 ~ ~~ ~~ ~~ ~ ~ ~(~x, ~y, ~z ) eK eimz ei(kx l y)dkdl , (5.4.18) F where F is the Froude number, as defined earlier. As discussed earlier, the linear theory holds for a large Froude number flow. On the other hand, for a small Froude number flow, nonlinear effects become more important and cannot be ignored. This will be discussed in the next subsection.
Equation (5.4.18) or (5.4.14) can also be solved numerically by applying a two- dimensional numerical FFT algorithm. Figure 5.17 shows an example of a linear, hydrostatic flow passing over a bell-shaped mountain with a Froude number of 100.
Near the surface, the pattern of vertical displacement resembles the surface topography,
(5.4.15), as required by the lower boundary condition. Slightly aloft from the surface at ~ z / 4 (Fig. 5.17a), a region of downward displacement forms a U-shaped disturbance
38
over the lee slopes of the mountain and extends some distance downstream. At a level further aloft, such as ~z (Fig. 5.17b), the region of downward displacement widens, moves upstream, and is replaced by a U-shaped pattern of upward displacement. The general upstream shift of downward and upward displacement is caused by the upstream phase tilt of upward propagating hydrostatic waves. At greater heights, the zone of disturbance continues to broaden, the disturbance directly in the lee of the mountain disappears, the patterns of upward and downward displacement become more wavelike, due to wave dispersion.
The U-shaped patterns of vertical displacements are explained by a group velocity argument (Smith 1980). The dispersion relation for internal gravity waves in a stagnant
Boussinesq fluid may be reduced from (3.6.10)
NK . (5.4.19) k2 l 2 m 2
With the hydrostatic approximation the above equation becomes
NK . (5.4.20) m
As discussed in Chapter 4, the energy propagation can be described by the group velocity components, which are
Nk Nl NK c ; c ; c (5.4.21) gx k mK gy l mK gz mm2
For steady-state waves on a basic flow, replacing by the intrinsic frequency Uk in
(5.4.20) leads to
NK m . (5.4.22) Uk
39
Adding U to cgx , the components of the group velocity in a frame fixed with the Earth become
Ul 2 Ukl Uk22 c ; c ; c . (5.4.23) gmx K 2 gmy K 2 gmz NK
In the coordinates fixed with the mountain or Earth, wave energy propagates from the energy source, i.e. the mountain, along straight lines with slopes
x c y c y c gmx ; gmy ; gmy . (5.4.24) zcgmz zcgmz xcgmx
The slope on the horizontal plane y / x may be evaluated from (5.4.23) and (5.4.24),
y k , (5.4.25) x l which is the geometric condition that the phase lines passing through the point (x, y) are radial lines from the origin. Using (5.4.23)-(5.4.24) again gives
Nxz y 2 . (5.4.26) UK
Since the mountain is the source of forcing, the horizontal wavenumber may be approximated by the mountain scale, i.e. Ka1/ , which yields
2 Nax yz . (5.4.27) U
Thus, the energy concentrates in a parabola or a U-shaped pattern at a certain height, as shown in Fig. 5.17.
In the above theory, the basic flow speed and Brunt-Vaisala frequency are assumed to be constant with height. In the real atmosphere, they normally vary with height. 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-
40
dimensional trapped lee waves is similar to that of Kelvin ship waves over the water surface. Figure 5.18 shows an example of the cloud streets associated with three- dimensional trapped lee waves produced by airflow past a mountainous island. The wave pattern is generally contained within a wedge with the apex at the mountain. The three- dimensional trapped lee waves are composed by transverse waves and diverging waves, as depicted in Fig. 5.19. 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. 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. Both of the transverse and diverging waves are mathematically associated with a stationary phase point, and the significant disturbance is confined within a wedge angle of about 19o28' with the x-axis.
5.4.2 Generation of lee vortices
The above linear theory of three-dimensional, stratified flow over mountains is valid only for high Froude number flow due to the limitations of the small-amplitude (linear) assumption. When the Froude number decreases, the perturbations generated by the mountain become larger and the flow becomes more nonlinear. Due to mathematical intractability, many observed phenomena associated with nonlinear flow over mountains, such as flow recirculation, stagnation points, flow splitting, and lee vortices, have been carried out in tank experiments and by nonlinear numerical simulations.
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a. Boundary layer separation
The flow pattern produced by laboratory tank experiments for three-dimensional, stratified flow with relatively large Froude numbers (e.g. F 2 ) past a bell-shaped mountain is similar to that predicted by linear theory as described in subsection 5.4.1.
Flow patterns are dramatically different for flow with smaller Froude numbers. Figure
5.20 shows a stratified flow with F 0.4 past an isolated mountain in a tank experiment.
The most eye-catching phenomenon is a pair of counter-rotating vortices formed in the lee of the obstacle. The formation of this pair of lee vortices is attributed to the boundary layer separation mechanism (Batchelor 1967; Hunt and Snyder 1980), as briefly summarized in the following. When the Reynolds number ( Re ) is sufficiently high
(where Re UL/ , U is the velocity scale, L length scale and kinematic viscosity), the boundary-layer flow develops a region of flow reversal near the surface due to an opposing pressure gradient in the direction of flow. The reversed flow meets the incoming flow and forms a stagnation point at which the streamline breaks away from the surface of the obstacle. This process is known as boundary layer separation.
Mathematically, the streamline of boundary layer separation is a line whose points are singular points of the solutions of the equations of motion in the boundary layer.
For three-dimensional, nonlinear, stratified viscous flow past a symmetric mountain, boundary layer separation first occurs on the center vertical plane before the mountain peak is reached. During the process, several singular points can form. Over the upslope on the center plane, an attachment point (node of attachment) Na forms, which forces part of the flow to recirculate back upstream along the upslope, where it meets the
42
incoming flow, and forms another stagnation point (saddle point of separation) Ss (Fig.
5.20a). Downstream of the obstacle on the center vertical plane, flow separates and forms a third stagnation point(s) (saddle points of attachment) Sa . The separated flow recirculates on this vertical plane, meets with the downslope flow and forms another saddle point of separation ( ) over the lee slope. On the surface (Fig. 5.20b), the recirculated flow from Na forces the incoming flow to split (i.e. flow splitting) at and part of the split flow recirculates and forms a pair of stationary lee vortices centered at the nodes of separation ( N s ). If the Froude number is decreased further, this flow pattern persists, but moves closer to the mountain peak and the lee vortices expand further downstream. Although an unrealistically large mountain slope of O(1), compared to that in the real world is often used in laboratory experiments, the simulated flow features are very similar to those observed in the real atmosphere.
b. Generation of lee vortices in an inviscid fluid
Using a nonlinear numerical model with free-slip lower boundary condition, a pair of counter-rotating vortices was found to form on the lee of an isolated mountain when a low-Froude number (e.g., F = 0.66, Fig. 5.21a), three-dimensional, stratified, uniform flow passes over the mountain (Smolarkiewicz and Rotunno 1989). The simulated results agree fairly well with laboratory tank experiments as shown in Fig. 5.20. The free-slip lower boundary condition implies no explicit surface friction is included in the model atmosphere. Although linear theory breaks down, at least locally, the vertical displacement field (Fig. 5.21c) still resembles the U-shaped pattern found in the linear theory described in subsection 5.4.1 (Fig. 5.17). A large-amplitude mountain wave
43
develops over the mountain peak (Fig. 5.21e). The trough of the vertically propagating gravity waves in Fig. 5.21e shifts upstream and becomes narrower, indicating a tendency toward collapse of the isentropic surfaces on the lee slopes of the mountain, which is also in agreement with the linear theory. Since the air parcels are able to flow almost directly across the mountain, this flow regime is characterized as the flow-over regime.
When the Froude number is reduced to approximately below 0.5, such as F 0.22
(Fig. 5.21b), a pair of counter-rotating vortices forms on the lee side and a saddle point of separation and a node of attachment are produced on the upstream side of the mountain, strikingly similar to the results obtained in laboratory experiments (Fig. 5.20b). The region of downward displacement is enlarged (Fig. 5.21d). The gravity wave response is drastically reduced, as much of the airflow is diverted around the flanks of the mountain and the disturbance appears to be much more horizontal (Fig. 5.21f). Below the mountain top, there is a recirculating flow associated with the lee vortices. This flow regime is characterized as the flow-around regime. Based on the nondimensional mountain height (or inverse Froude number - Nh/ U ) and horizontal mountain aspect ratio ( ba/ ), four classes of wave and flow phenomena of importance in three- dimensional, stratified, uniform, hydrostatic flow past an isolated mountain can be identified (Fig. 5.22): (1) linear mountain waves, (2) wave breaking, (3) flow splitting, and (4) lee vortices.
The key question of the numerically simulated lee vortices as shown in Fig. 5.21 is the source of vorticity. In the absence of surface friction, boundary layer separation will not occur and thus cannot be held responsible for the formation of the lee vortices.
Although many detailed dynamics of this problem are still topics of current research, the
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basic dynamics for the generation of lee vortices can be understood through the following two major theories: (1) tilting of baroclinically-generated vorticity (Smolarkiewicz and
Rotunno 1989) and (2) generation of internal potential vorticity by turbulence dissipation in numerical simulations (Smith 1989b; Schär and Smith 1993a, b).
c. Tilting of baroclinically-generated vorticity
The mechanism of baroclinically-generated vorticity tilting can be understood by taking cross differentiations of (2.2.1) – (2.2.3) to yield the inviscid vorticity equation
x p V ( ) V , (5.4.28) t 2
where x V ( , , ) is the three-dimensional vorticity vector. The last term on the right side of the above equation represents the generation of vorticity by baroclinicity.
Once local vorticity anomalies are generated, they are advected by the flow field through the first term or tilted and stretched through the second term on the right side of (5.4.28).
For mountains with small aspect ratio of the obstacle height and horizontal width, the baroclinicity term reduces to (e.g., Epifanio 2003):
∇∇ x p kb x ∇ , (5.4.29) 2 where b is the buoyancy.
Figure 5.23 shows a schematic diagram depicting the generation of leeside vorticity by the vertical tilting of baroclinically generated horizontal vorticity. A negative x -
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vorticity, 0 , is generated on the right upslope baroclinically by the relatively cold air along the center line and the relatively warm air to the right (facing downstream), as indicated by (5.4.29). This negative x -vorticity is then swept downstream and produces a positive vertical vorticity, 0 , to the lee by the vertical tilting of the x-vorticity, as implied by (5.4.28). Similarly, a positive -vorticity anomaly generated over the left upslope is tilted into a negative vertical vorticity to the lee. As these vertical vorticity anomalies intensify, recirculating warm-core eddies develop as a result of reconnection.
This mechanism dominates during the rapid start-up, early stage, over a nondimensional time Ut/ a = O(1), in which the flow is essentially inviscid and adiabatic and the potential vorticity (PV) is conserved (Schar and Durran 1997).
d. Generation of potential vorticity by turbulence dissipation
At a later stage, the associated thermal anomalies generated by baroclinicity are eroded by dissipative and diffusive processes, whereby the warm surface anomalies are converted into PV. During this stage, the flow is controlled by dissipation and is accompanied by the PV generation over a nondimensional time of O(10) – O(100) (
and Durran 1997). Note that the conservation of potential vorticity is violated in regions of flow stagnation, such as in the region of upstream blocking where the isentropic surface intersects the ground, and the region of wave breaking above the lee slope where turbulence occurs (Fig. 5.24a). The dynamics of dissipative generation of PV is directly linked to the reduction in the Bernoulli function within the wake, as demonstrated in steady shallow-water flow past an obstacle (Schär and Smith 1993a).
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The shallow-water theory can be extended to stratified fluid flow by considering the PV
(q) which satisfies a conservative equation of the form (Haynes and McIntyre 1990):
()q J 0 , (5.4.30) t where q is defined as
q α , (5.4.31) and the total PV flux (J) is given by
J =qQ V a F x . (5.4.32)
In the above equation, Q ( D / Dt ) is the diabatic heating, a the three-dimensional absolute vorticity vector, and F the viscous force per unit mass. In this section, we have
assumed that the Earth rotation is negligible thus α .
It can be shown that
V J= x Β , (5.4.33) tt where
BVV /2 cp T gz (5.4.34)
is the Bernoulli function. In a steady state flow, the Bernoulli function is conserved following the flow. In addition, (5.4.33) reduces to
J = x Β = n x Β , (5.4.35) n where n is a unit vector oriented perpendicular to the isentropic surface and pointing toward warm air. The generalized Bernoulli’s theorem (Schär 1993), (5.4.35), indicates
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that non-zero PV fluxes must be present where there is a variation in the Bernoulli function along any isentropic surface. Figure 5.24b shows a schematic of PV generation by turbulence dissipation on an isentropic surface in steady-state stratified flow past an isolated mountain. The narrow dissipative region may be produced by turbulence associated with wave breaking, a hydraulic jump or blocking, and generates Bernoulli function deficit in the wake extending downstream. Based on (5.4.34), PV is generated in the dissipative region and advected downstream along the edge of the wake. A pair of counter-rotating vortices may form in the wake if the vertical vorticity associated with the generated PV is sufficiently strong.
It appears that the above PV analysis is able to explain the close relationship between dissipative turbulence and PV generation for a low-Froude number, stratified flow over an isolated mountain. The causality, however, is still unclear due to the steady state assumption. In addition, an assumption of balance is required in order to infer the structure of the flow from the distribution of PV (Hoskins et al. 1985). In the near field of the wake, these balance constraints are constantly strongly violated due to the presence of the strong surface temperature gradient over the lee slope, which results from the upstream blocking (Epifanio and Rotunno 2005). Therefore, although the PV generation may have important implications on the downstream evolution of orographic wakes and lee vortices, a fundamental understanding of the wake formation is needed.
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
48
process is also known as vortex shedding. Any noise, impulsive disturbance, or asymmetric forcing in the wake flow can trigger an instability, which gives way to a vortex street or vortex shedding. Figure 5.25 shows an example of a von Kármán vortex street formed in the atmosphere to the lee of a mountainous island. The von Kármán vortex street or vortex shedding has also been simulated by many nonlinear numerical models, such as that shown in Fig. 5.28a.
5.5 Flows over larger mesoscale mountains
5.5.1 Rotational effects
In the previous sections, effects of Earth’s rotation are neglected. This is
approximately valid for flow with Rossby number ( Ro U/ fL , where L is the horizontal scale of the mountain) much larger than 1. However, for flow over mountains with
Ro O(1) or smaller, the effects of Earth’s rotation cannot be ignored. In this situation, the advection time for an air parcel to pass over the mountain is too large to be ignored compared to the period of inertial oscillation due to Earth’s rotation ( 2 / f ). Flow past many mesoscale mountain ranges, such as the European Alps, US Rockies, Canadian
Rockies, Andes, the Scadinavian mountain range, the New Zealand Alps, and the Central
Mountain Range of Taiwan, belong to this category. For flow over a very broad large- scale mountain, which gives a very small Rossby number, the flow can be approximated by the quasi-geostrophic theory. In general, however, for flow over mesoscale mountains, the Rossby numbers are on the order of 1, for which the influence of the Coriolis force is too large to be ignored, but too small to be approximated by the quasi-geostrophic theory.
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In considering the small-amplitude, two-dimensional ( /y 0 ), steady state, inviscid, rotating, uniformly stratified, Boussinesq flow over an infinitely long ridge on an f -plane, the governing equations can be derived from (3.6.1)-(3.6.5),
2 2 2 2 2 2 w'''' w 2 w 2 w U2 2 2 f 2 N 2 0. (5.5.1) x x z z x
Similar to the procedure for obtaining solutions described in subsection 5.2.1, the vertical displacement can be obtained by,
(x, z) 2 Re hˆ(k)ei(kxmz )dk , (5.5.2) 0 where hˆ(k) is the one-sided Fourier transform of the mountain shape h(x) (Appendix
5.1)
1 hˆ( k ) h ( x ) eikx dx , (5.5.3) 2 and m is the vertical wave number
k 2 (N 2 k 2U 2 ) m . (5.5.4) k 2U 2 f 2
Note that (5.5.4) is similar to that defined in (3.6.10), in which kU for a time- dependent problem. Several regimes exist for the mountain waves in a rotating stratified atmosphere, similar to those of inertial gravity waves discussed in Chapter 4 and summarized in Table 3.2. We have already discussed two flow regimes in earlier
2 2 2 2 sections, such as the nonrotating evanescent flow regime for k U N f , and the the nonrotating vertically propagating wave regime for N 2 k 2U 2 f 2 .
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To demonstrate the rotational effects, we consider an extreme case of a flow past a very broad mountain ridge. Assuming a bell-shaped mountain range, (5.2.5), its Fourier
ˆ ka transform can be obtained: h(k) (hm / 2)ae if k 0 [(5.2.6)]. In this situation, we have N 2 f 2 k 2U 2 and the vertical wave number reduces to m ikN / f . As discussed in Chapter 4, this is equivalent to making the quasi-geostrophic approximation.
ˆ Substituting h(k) and m into (5.5.2) leads to the solution for a quasi-geostrophic flow over a very broad bell-shaped mountain ridge,
h a2 ( Nz / fa 1) (,)xz m . (5.5.5) x2 a 2( Nz / fa 1) 2
The above solution can be extended to quasi-geostrophic flow over a three- dimensional, isolated mountain, as shown in Fig. 5.26a. The lifting of surfaces aloft is less than the mountain height, but the lifting is more widespread. Assuming a non- rigid lid top boundary, the air column upstream of the mountain is stretched slightly, producing weak cyclonic vorticity. Over the mountain, the air column is shortened producing anticyclonic vorticity. This anticyclonic vorticity is associated with a mountain-induced high pressure or anticyclone. On the lee side, the air column is slightly stretched again, producing weak cyclonic vorticity. Figure 5.26b shows the streamline pattern near the surface on a horizontal plane for the flow associated with Fig. 5.26a.
When a straight incoming flow approaches a mountain ridge in the Northern Hemisphere, it turns cyclonically (left facing downstream) slightly upstream of the mountain, anticyclonically over the mountain, and cyclonically slightly on the immediate lee side in response to the slight vorticity stretching, major vorticity shrinking, and slight vorticity stretching of the vertical vorticity, in order to conserve the potential vorticity (Fig. 5.26a).
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This gives the streamline pattern as depicted in Fig. 5.26b. The perturbation velocity and pressure field decay away from the mountain. The upwarping of isentropic surfaces and the cold-core anticyclone near the ground, caused by a cold air mass at the surface, is analogous to a mountain anticyclone. When a preexisting cyclone passes over a mountain, its low pressure or cyclonic vorticity is weakened by the mountain-induced high pressure or even completely suppressed the low (filled in) on a weather map. After it passes over the mountain, the cyclone reappears on the lee side. In addition, the cyclone track is often deflected by the mountain-induced anticyclone. This will be addressed in the next subsection.
Due to mathematical intractability, asymptotic methods or numerical techniques are often employed to solve (5.5.2). Figure 5.27 shows an asymptotic solution of (5.5.2)
multiplied by the non-Boussinesq factor s /()z . This flow belongs to an intermediate or mesoscale mountain wave regime between the nonrotating flow regime
(e.g., Fig. 5.3b) and the quasi-geostrophic flow regime (e.g., Fig. 5.26a). In this intermediate regime with U / a f , both the inertia force and Coriolis force play comparable roles in generating and maintaining the mesoscale mountain waves. The influence of the Coriolis force is evident in the lateral deflection of the streamlines and in the dispersive tail of the longer waves trailing behind the mountain at upper levels. The
horizontal wavelength is Lf 2 U / f 600 km near the surface and the vertical
wavelength is z 2U / N 6.28 km . Unlike nonrotating flows, the upstream blocked decelerated zone produced by a mountain ridge on a uniform, stratified flow on an f plane does not propagate to infinity (Pierrehumbert and Wyman 1985). Instead, it attains a maximum extent on the order of the Rossby radius of deformation ( Nh/ f ).
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Some rotational effects on three-dimensional stratified airflow past an isolated, mesoscale mountain are illustrated in Fig. 5.28. With rotation, a significant portion of the upstream fluid particles move leftward (facing downstream) in response to the excessive pressure gradient force versus the reduced Coriolis force because the weakened easterly flow is out of balance geostrophically (Fig. 5.28a), compared with the case without rotation (Fig. 5.28c). The separation point is shifted to the right-hand side in the rotating flow (Fig. 5.28a), instead of located along the centerline ( y / a 0 ) (Fig. 5.28c).
Downstream of the mountain, vortex shedding occurs. Over the mountain peak, the hydrostatic mountain wave is very weak because most of the flow is blocked by the high mountain and is deflected to the right or left, instead of climbing up and passing over the mountain (Fig. 5.28b). On the cross section along the centerline ( ), both rotating and nonrotating flow produce blocking, high pressure and cold air over the upslope and low pressure and warm air over the downslope (Figs. 5.28b and 5.28d). A southerly barrier jet forms along the upslope through geostrophic adjustment (see Ch. 1) in response to this reversed (against the basic flow) pressure gradient force in a rotating atmosphere (Fig. 5.28a). In contrast, there is no barrier jet in a nonrotating flow (Fig.
5.28c). Due to the strong shear flow passing over the left tip of the mountain, a jet forms through the corner effect caused by the Bernoulli effect (Figs. 5.28a and 5.28c). In the absence of rotation (Figs. 5.28c), a pair of cyclonic and anticylonic vortices is generated on the lee, but these are symmetric with respect to the centerline. The vortex shedding in a rotating flow also produces pressure and potential temperature perturbations far downstream (Fig. 5.28b). Three-dimensional flow past an isolated mountain in a rotating frame of reference has also been simulated using many other nonlinear numerical models.
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In addition to the above rotational effects, a topographic Rossby wave can be produced for a flow over a mesoscale mountain. Consider an air column in a barotropic atmosphere that is pushed up northward against a sloping surface of a east-west oriented mountain range. The air column tends to aquire an anticyclonic vorticity ( < 0) in order to conserve the potential vorticity ( ()/ fH), where H is the effective depth of the air column, due to decreasing H. An air parcel in the air column then turns anticyclonically, assuming f remains constant and the surface friction is negligible, and moves downslope. This forces the air column to stretch, thus acquiring cyclonic vorticity due to increasing H, and turn upslope. This wavelike motion is called a topographic
Rossby wave because the sloping surface plays an analogous role to that of the effect.
5.5.2 Lee cyclogenesis
The lee sides of mesoscale or large-scale mountains, such as the Alps, Rocky
Mountains, the East Asian mountains and the Andes, are favorable regions of cyclogenesis. This type of cyclogenesis is known as lee cyclogenesis, and can be defined as the formation of a cyclone with strong positive vertical vorticity or an appreciable fall in pressure with a closed circulation formed in the lee of a mountain that then drifts away.
Figure 5.29 shows the cyclogenesis frequency in the Northern Hemisphere during wintertime from 1899 to 1939. Frequency maxima in the figure are located to the lee of major mountain ranges, such as south of the Alps over the Gulf of Genoa and to the east of the Rockies.
Similar to diabatic heating, major mountain ranges serve as sources of stationary wave trains, which produce strong baroclinic zones favorable for storm tracks (Blackmon
54
et al. 1977). Two well-known storm tracks, the Pacific trough and the Atlantic trough, are associated with stationary, Rossby wave trains emanating from the Himalayas and
Rockies, respectively. Based on the above definition of lee cyclogenesis, the weak stationary low (pressure depression) associated with these stationary Rossby wave trains in the lee of mountains should be distinguished from the true lee cyclogenesis (Chung et al. 1976). Observations indicate that lee cyclogenesis may be influenced by different factors, such as the synoptic conditions, as well as the orientation and geometry of the mountain range to the prevailing wind. For example, the Alps are more isolated and parallel to the baroclinic flow, while the Rockies are more elongated and normal to the baroclinic flow. In the following, we will focus the discussion on the theories of lee cyclogenesis over two major mesoscale mountain ranges, the Alps and Rockies. These theories can be applied to lee cyclogenesis over a number of mountain ranges in other parts of the world, depending upon the synoptic conditions and terrain geometry.
a. Alpine lee cyclogenesis
As depicted in Fig. 5.29, a frequency maximum of lee cyclogenesis can be found to the south of the Alps over the Mediterranean Sea. Lee cyclogenesis over the Alps has been studied extensively, compared to that over other mountain ranges, partially due to the Alpine Experiment (ALPEX) field project held in 1982. The general properties (=> characteristics) of the Alpine lee cyclogenesis can be summaried as follows (e.g., Tibaldi et al. 1990):
(a) Lee cyclogenesis often occurs in association with a preexisting synoptic-scale
trough or cyclone that interacts with the orography;
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(b) The development of the lee cyclone starts before the strong thermal contrasts
associated with cold frontal penetration take place in the lee;
(c) The scales of mature and deep lee cyclones are on the order of the Rossby radius
of deformation ( NLz / f , where Lz is the vertical scale of motion). The influence
of orography takes the form of a high-low dipole, which has the same horizontal
and vertical scales of the cyclone; and
(d) A two-phase deepening process is found. In the first phase, a cyclone deepens
very rapidly, but is shallow. In the second phase, the cyclone develops less
rapidly, but extends through the whole troposphere with horizontal scales
comparable to the Rossby radius of deformation.
Based on upper-level flow, two types of lee cyclogenesis over the Alps can be identified: the southwesterly flow (Vorderseiten) type and the northwesterly flow (Überströmungs) type (Pichler and Steinacker 1987). Both types are accompanied by blocking and splitting of low-level cold air by the Alps. One part of the low-level cold air flows anticyclonically around the Eastern Alps, often leading to a bora (bura) event over the Adriatic Sea, while another part of the cold air flows around the Alps cyclonically and leads to the mistral wind, blowing through the Rhone Valley towards the Gulf of Lyons (to the south of France). For the southwesterly type (Fig. 5.30a), cyclogenesis starts with the advance of an eastward moving trough, where a wave at the corresponding surface cold front always forms in about the same place, i.e. the Gulf of Genoa (to the south of Italy and north of Corsica), due to the blocking and flow splitting effect of the Alps. A warm front is often generated over northern Italy. For the northwesterly type (Fig. 5.30b), the upper-level flow blows generally from the northwest, which crosses the Alps and generates a cyclone on the lee. The wave formation in the surface cold front occurs predominantly in the Gulf of Genoa, the same location as for the southwesterly type, due to the shape of the Alpine mountains. A short wave trough
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embedded in the general northwesterly flow can, however, produce an intermediate southerly flow component. In the northwesterly type of lee cyclogenesis, there is no new cyclone formed in the lee of the Alps. Usually, there is a preexisting “parent” cyclone upstream northwest of the Alps, which is weakened by the anticyclone on the windward side of the Alps and intensified in the lee in a favorable baroclinic flow. Two major theories for Alpine lee cyclogenesis have been proposed in the past: (1) baroclinic lee waves (Smith 1984) and (2) orographic modification of baroclinic instability (Speranza et al. 1985). As can be seen from the following description, it appears that the first mechanism is more applicable to the southwesterly type, while the second mechanism is more applicable to the northwesterly type.
The basic dynamics of the baroclinic lee wave theory can be understood by considering a quasi-geostrophic flow with an idealized, linear shear over an isolated mountain. The basic wind represents an incoming cold front or baroclinic zone (with cold air to the west), similar to the southwesterly type (Fig. 5.30a). This basic wind profile allows for the growth of a standing baroclinic wave in the lee (south) of the mountain. Although a baroclinic wave is often used to imply a growing, baroclinically unstable wave, here we define it more generally as a surface-trapped wave in a baroclinic current whose restoring force is associated with temperature advection at the boundary.
Based on the quasi-geostrophic approximation, the geostrophic, Boussinesq potential vorticity equation on an f-plane can be written as (Smith 1984):
2 Dg f 2P P 0, (5.5.6) 2 zz Dt N where P is the pressure, the terms inside the bracket are the potential vorticity,
Dg//// Dt t u g x v g y , ug Py /(o f ) , vg Px /(o f ) , and the thermodynamic equation at the surface is,
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D g w 0 , at z 0, (5.5.7) Dt z
where is the potential temperature. Assuming ugg U()' z u and vgg V()' z v and the basic flow velocity is a linear function of height, i.e. U() z Usz U z and
V() z Vsz V z , then the potential vorticity in (5.5.6) vanishes everywhere. Here Uo and
Vo are the basic wind speeds at surface. Thus, any perturbation to this basic state must obey,
2 2 f p' p'zz 0 (5.5.8) N 2 and
p''' p p 2 Us V s U z V z o N w'0 at z 0, (5.5.9) t x y z x y
2 where N( g /o )( / z ) . The above two equations form a closed system as long as w' is known at z 0 , which can be specified by the linear lower boundary condition, w'( x , y , z 0) U h / x V h / y h(x, y) ss, where is the mountain height function.
If one considers a two-dimensional, steady-state flow over a bell-shaped mountain ridge, h(x) ha2 /(x2 a2 ) , an analytical solution can be obtained by applying the
Fourier transform to both (5.5.8) and (5.5.9) and an upper boundedness boundary condition (which differs from the rigid lid boundary condition) in a semi-infinite vertical plane. The solution can be written as
z/ H ikx ˆ 2 h() k e e p'( x , z ) os U N 2Re dk , (5.5.10) 0 UHUsz/
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where H (k) f / Nk and hˆ(k) (ha / 2)ekx. In deriving (5.5.10), we have applied the one-sided Fourier transform. According to the Riemann-Lebesgue Lemma (Appendix
5.1), the integral will go to 0 as x due to the rapid oscillation of the term eikx . In this case, the airflow will be disturbed only near the mountain. However, if the denominator of the integrand vanishes for some value of k , the integral will not vanish.
This happens when there is a back shear. For example, when U s 0 , this requires
** ** * U z 0 and the basic wind vanishes at height H f/ Nk (where k NH/ f and H is the wind reversal level). Thus, the vanishing of the basic wind U (z) at some height is the condition for obtaining a disturbance away from the mountain. Physically, this is an orographically-forced standing baroclinic wave which has a zero phase speed. Evaluating (5.5.10) gives p'(x, z) 0 , for x 0 and (5.5.11a)
ˆ * z / H* * p'(x, z) (4oNf )h(k )e sin k x for x 0. (5.5.11b)
Equation (5.5.11b) describes a train of standing baroclinic lee waves on the lee of the mountain. With typical flow and orographic parameters for Alpine lee cyclogenesis, such
-3 -1 4 -1 3 5 as o 1 kg m , N 0.01 s , f 10 s , h 3x10 m , a 2.5x10 m ,
H *3 5x10 m , and k* f/( NH * ) 2x10 6 m -1 , a pressure perturbation p ' 28 hPa can be generated. The wavelength of the baroclinic lee wave is 2 /k * 3000 km , which is comparable to observed baroclinic waves over the Alps.
To obtain the transient solution, we take the Fourier transform of (5.5.8), applying the upper boundedness boundary condition, and then substitute the solution to (5.5.9) with the linear lower boundary condition. This leads to
Hg ikU hˆ pˆ() t AeBt o z s , (5.5.12) o B where
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B ik() Usz HU , (5.5.13) and A is a complex coefficient to be determined by the initial condition. The solution in physical space can then be obtained by taking the inverse Fourier transform of (5.5.12).
One can apply a FFT algorithm to obtain the numerical solution. Figure 5.31 shows examples of the steady-state solution, (5.5.11), and the transient solution of (5.5.12) with an initially undisturbed flow, p'(t 0) 0 . This leads to rapid lee cyclogenesis with pressure tendencies from 4 to 6 mb/3h. The fluid is, in effect, trying to form the first trough of a standing baroclinic lee wave. Compared with the steady state solution, the transient solution produces the simple pattern of a wave growing away from its source.
The above system has been extended to obtain the following (Smith 1986): (a) three- dimensional numerical solutions for Alpine cyclogenesis, which show qualitative agreement with observations; (b) an upshear phase tilt with height which indicates that the growing baroclinic wave is able to extract energy from the available potential energy stored in the baroclinic shear flow; (c) with a rigid lid added to the model, there is a rapid two-phase lee cyclone growth supported by baroclinic instability of the Eady type (1949);
(d) the baroclinic lee waves belong to the regime of mixed inertia-gravity waves and trapped baroclinic lee waves (Sec. 3.8), thus making the theory more applicable to flow with moderate Ro and moderate Ri , or small Ro and moderate ; and (e) the basic features predicted by the linear theory are consistent with those simulated by nonlinear numerical models (Lin and Perkey 1989; Schär 1990).
In the theory of orographic modification of baroclinic instability for Alpine lee cyclogenesis, the formation of lee cyclones is attributed to the baroclinically unstable modes modified by orography. To derive the governing equations, we assume a small-
60
amplitude mountain and a basic flow with constant stratification and vertical shear, and a
zonal wind that vanishes at z 0, bounded between and z H in the vertical and
by lateral walls at y Ly / 2 . For this type of flow, the linear, nondimensional
governing equations and boundary conditions can be written as (Speranza et al. 1985):
2 zz 0 ,
zt x J yz(,/) h R o at ,
zt x xz 0 at z 1, and
0 at , (5.5.14)
where is the streamfunction, J the Jacobian [defined as J yz(,) y z y z ], h
the mountain height, and Ro the Rossby number. Note that (5.5.14) is equivalent to
(5.5.8) of the baroclinic lee wave theory except that the streamfunction is used. Assuming
(t, x, y, z) (x, y, z)eit and applying the Fourier series expansion to the above
system reduces it to an eigenvalue problem, which allows one to find the most unstable
baroclinic wave mode from the eigenvalues. Figure 5.32 shows one example of Alpine
lee cyclogenesis based on the above theory. The disturbance at the surface (Figs. 5.32a
and 5.32b) is more localized near the mountain than that at the mid-troposphere (Figs.
5.32c and 5.32d). The total streamfunction fields (Figs. 5.32b and 5.32d) appear to be
able to capture some basic structures of Alpine lee cyclones in the mature stage. Lee
cyclogenesis is explained as the intensification of the incident baroclinic wave taking
place on the southern (warm) side of the mountain, together with a weakening of the
wave amplitude on the northern (cold) side.
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When a vertical wall is used to represent the Alps and to block all meridional flow below the crest level, a lee cyclone develops through baroclinic instability (Egger 1972;
Pierrehumbert 1986), as in Fig. 5.32, and so the situation illustrated in Fig. 5.32 may thus be considered to belong to the mechanism of orographic modification of baroclinic instability. In fact, the above two theories for Alpine lee cyclogenesis are related because both of them are built upon the baroclinic instability theory of Eady (1949), as can be seen from (5.5.8) and (5.5.14), but use different initial and upper boundary conditions and basic wind profiles. In the baroclinic lee wave theory, an undisturbed flow or a localized baroclinic wave is used to initiate the process, while a fully-developed baroclinic wave is used as the initial condition in the theory of orographic modification of baroclinic instability. In addition, the theory of orographic modification of baroclinic instability uses a rigid lid upper boundary condition, as in Eady’s model, while the baroclinic lee wave theory does not impose a rigid lid but uses a back-sheared north-south basic wind profile. Based on the above discussions, it appears that the baroclinic lee wave mechanism is more applicable to the southwesterly cyclogenesis type, while the mechanism of orographic modification of baroclinic instability is more applicable to the northwesterly type which normally contains a preexisiting cyclone.
Some other physical processes may also play important roles in the Alpine lee cyclogenesis, such as nonlinear processes (Tafferner and Egger 1990), geostrophic adjustment to upper-level potential vorticity advection (Bleck and Mattock 1984), and orographic modification of the baroclinic parent cyclone (Orlanski and Gross 1994), under certain synoptic situations. The two-phase deepening process has been attributed to the rapid formation of a low-level orographic vortex, followed by its baroclinic and
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diabatic interaction with an approaching upper-level trough (Aebischer and Schar 1998).
In the first phase, orographically generated elongated bands of positive and negative PV, i.e. PV banners or streamers, may wrap up and subsequently contribute to the low-level
PV anomaly within the developing cyclone. The second phase is interpreted as mere orographic modification of baroclinic instability.
b. Rockies lee cyclogenesis
The major difference that sets apart the Rockies lee cyclogenesis from the Alpine lee cyclogenesis is that the Rocky Mountains are elongated in a direction normal to the basic wind, while the Alps are elongated in a direction parallel to the thermal wind (basic wind shear). In addition, the Rocky Mountains have a much larger horizontal scale than the
Alps in the direction of the low-level flow. Three stages can be identified for the lee cyclogenesis over the Rocky Mountain cordillera:
(a) Stage I: A parent cyclone approaches the Rocky Mountain cordillera from the
northwest. As the cyclone moves closer to the mountains, it decelerates and
curves to the north.
(b) Stage II: During the passage of the parent cyclone over the cordillera, the low-
level portion of the cyclone weakens or fills in (disappears) over the mountain,
while the upper-level trough continues to move over the mountain, but with some
track deflection;
(c) Stage III: A new surface cyclone forms in the lee of the mountain range and to the
south of the original track. In the meantime, an upper-level trough, which is
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associated with the parent cyclone, has progressed over the mountain and helps
strengthen the surface cyclone through baroclinic development.
Figure 5.33 shows an example of cyclogenesis which occurred in the lee of the U.S.
Rocky Mountain cordillera on 18-20 March 1994. At 1200 UTC 18 March (3/18/12Z), a parent cyclone was located over the southern Oregon and was approaching the Rockies. It subsequently moved eastward over southern Idaho by 3/19/00Z (Stage I). Stage II occurs between 3/19/00Z and 3/19/12Z. At 3/19/12Z, a new cyclone forms over eastern
Colorado, which is to the southeast of the incidence of the parent cyclone. As the upper- level trough begins moving east of the mountains, the phase shift between the upper trough and the lee cyclone begins to decrease and classical baroclinic development takes place.
Similar to Alpine lee cyclogenesis, the Rockies lee cyclogenesis has also been studied extensively in the past and several mechanisms have been proposed. Due to the fact that a parent cyclone is almost a prerequisite for the Rockies lee cyclogenesis and that the flow is highly baroclinic, the formation of lee cyclones can be reasonably explained by the theory of orographic modification of baroclinic instability, described by (5.5.14). The major difference is that the low-level tracks of the Rocky Mountains cyclones are strongly influenced by the mountain geometry, as will be discussed in subsection 5.5.3, and by the upper-level trough's effect on the intensity of the lee cyclone.
The theory of orographic modification of baroclinic instability, as described in
(5.5.14), can be extended to avoid the limitations of gentle sloping and small amplitude mountains by allowing for the interaction of growing normal modes in a baroclinic channel with isolated topography of finite height (Buzzi et al. 1987). In addition, the
64
isolated mountain may be elongated in the north-south direction to mimic the Rockies topography. Figure 5.34 shows the time evolution of the fastest growing baroclinic mode in the lower layer of the two-layer channel model. The three stages summarized earlier are reasonably reproduced by this linear stability model. The track deflection will be explained in Subsection 5.5.3.
Note that nonlinearity should be included in the above lee cyclogenesis theory when applied to the real atmosphere (Tafferner and Egger 1990). Various physical processes have been emphasized in other theories of the orographic modification of baroclinic instability mechanism, such as: (1) superposition of a growing baroclinic wave and a steady mountain wave (e.g., Hayes et al. 1993), and (2) Eady edge waves, i.e., an orographically modified surface potential temperature distribution leading to the upstream northward deviation of the baroclinic waves (Davis 1997).
c. Mesoscale lee cyclogenesis
A mesoscale vortex formed on the lee of a mesoscale mountain, as discussed in the subsection 5.4.2, can further develop into a lee cyclone by aquiring low pressure at the center of the vortex through various processes, such as latent heating associated with moist convection, surface heating over a warm sea surface, merging with a nearby mesolow, and interaction with an approaching upper-level trough. This type of mesoscale cyclogenesis has been observed in the lee of some mesoscale mountains and simulated by numerical models. Examples are the Denver cyclone (Crook et al. 1990), the mesocyclones to the east of Taiwan’s CMR (Sun et al. 1991; Lin et al. 1992) , and the lee
65
cyclones generated by the northern Korean mountain complex over the East (Japan) Sea
(Lee et al. 1998).
5.5.3 Orographic influence on cyclone track
Similar to what is shown in Fig. 5.33, track deflection of cyclones by mountains also occurs in other mesoscale mountain ranges, such as the Appalachians and Greenland, as long as the mountain range has appreciable extension in the direction normal to the prevailing wind in which the cyclone is embedded. Orographic influence on cyclone tracks has also been observed for tropical cyclones passing over mesoscale mountains, such as the northern Philippines and the Caribbean Islands, the Sierra Madre of Mexico, and Taiwan. Figure 5.35 shows examples of track deflection of tropical cyclones past the
CMR of Taiwan. Depending upon the intensity, steering wind speed, size of the cyclone, landfalling location and impinging angle, the tracks over CMR can be classified as continuous or discontinuous. Figure 5.36 illustrates the orographic influence on cyclone track through the turning of the basic flow and the outer circulation of the cyclone. The orography is assumed to be shallow, so that the blocking is relatively weak and most of the low-level air is able to pass over the mountain. The track deflection of a cyclone embedded in a straight incoming flow on a mesoscale mountain ridge, as shown in Fig.
5.36a, can be explained by either the force balance or the conservation of potential vorticity, as discussed earlier along with Fig. 5.26a.
Cyclone tracks can also be affected by the orographic influence on the outer circulation of the cyclone. This process is illustrated in Fig. 5.36b. Air ascends in the southwest and northeast quadrants and descends in the northwest and southeast quadrants
66
with respect to the intersection of the original (west-east) undisturbed track and the mountain crest. This produces a positive local vorticity tendency ( /0 t ) in the northwest and southeast quadrants due to vorticity stretching, ()/ f w z (e.g., part of the second term on the right side of (5.4.28) with the Coriolis term added). Similarly,
/0 t is produced in the southwest and northeast corners due to vorticity shrinking.
This helps migrate the cyclone toward the northwest corner and generate a secondary
(new) cyclone in the southeast quadrant. The lee cyclone develops quickly once it is coupled with the upper portion of the parent cyclone or trough. Note that this process will produce a lee cyclone that is at most of the same strength as the parent cyclone. In order to generate a stronger lee cyclone, other processes are required, such as barolinic instability or upstream flow blocking.
The continuity of cyclone tracks and degree of track deflection is directly related to the degree of orographic blocking. Figure 5.37 shows orographic influence on cyclone track deflection for a passive cyclone in a uniform flow past an isolated, finite-length mountain on an f-plane. With weak blocking (panel a), the cyclone is deflected to the right (facing downstream) and a weak secondary cyclone forms on the lee. The track is continuous in this case. With strong blocking (panel b), the cyclone is deflected to the left and a relatively strong secondary cyclone forms on the lee. In this case, the track is discontinuous. It is found that the track deflection is associated with the local vorticity tendency ( / t ) which is dominated by the vorticity advection and vorticity stretching while the cyclone is crossing over a mesoscale mountain range (Lin et al. 1999; 2005).
For example, for the case of Fig. 37a (weak blocking), the track deflection is dominated by vorticity advection. For the case of Fig. 37b, strong blocking tends to generate a
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strong northerly jet along the upwind slope of the mountain range, which, in turn, produces a local vorticity tendency to the left (north) of the undisturbed cyclone track by vorticity advection. The strong secondary vortex on the lee is generated by the vorticity stretching. The degree of orographic blocking can be measured by nondimensional control parameters, such as the basic-flow Froude number (U/ Nh ), the vortex Froude
number (VT / Nh ), and the relative scale of the cyclone and the orography in the direction
perpendicular to the steering flow ( RL/ y ), where U is the basic flow speed, VT the
characteristic tangential wind speed of the parent cyclone (such as Vmax ), N the Brunt-
Vaisala frequency, h the mountain height, and R is the cyclone scale represented by its radius of maximum tangential wind.
5.6 Other orographic effects
In addition to the phenomena discussed in previous sections, orography also plays a significant role in influencing the general airflow and weather systems, such as frontal passage, the formation of coastally trapped disturbance, cold-air damming, gap flow, orographic precipitation and foehn wind. The dynamics of orographic precipitation and foehn wind require backgrounds in thermally forced flow, mesoscale instabilities, and moist convection. Descriptions of these are made in Chapter 11. The orographic effects on other phenomena are briefly described next.
5.6.1 Effects on frontal passage
When a front passes over a mesoscale mountain range, it is often distorted by the mountain. As soon as the front impinges on the mountain, an anticyclonic circulation,
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associated with the mountain-induced high pressure, is produced in the cold postfrontal air and the front advances faster on the left (facing the frontal movement). Figure 5.38 shows a conceptual model of a two-dimensional front passing over a mesoscale mountain ridge. The mountain blocks and decelerates the approaching front at the surface while the upper-level potential vorticity anomaly associated with the front is affected to a much lesser degree as it moves across the mountain. When the blocking is strong, the surface front remains trapped on the windward slope and the frontal propagation is discontinuous across the ridge. During frontal passage over a mountain, several interesting phenomena can occur, such as the development of a lee trough or of a secondary trough along the lee slope, the separation of the upper-level and lower-level frontal waves, and the coupling of the upper-level frontal wave with the secondary trough in the lee of the mountain.
The passage of three-dimensional fronts over mesoscale mountains is more complicated than that of a two-dimensional front. Figure 5.39a shows a schematic of a cold front passing over the Alps, in which the northeastern portion of the front accelerated eastward, while the southwestern portion of the front was retarded at least initially. In other words, the front passed over the mountain anticyclonically. This type of deformation of the surface fronts has also been observed for Mei-Yu fronts passing over the Central Mountain Range (CMR) of Taiwan (Fig. 5.39b). The Mei-Yu (Baiu or
Changma) front is a nearly stationary, east–west-oriented weak baroclinic zone in the lower troposphere occurring from mid- to late spring through early to midsummer, and typically stretching from eastern China to Taiwan, Japan or Korea. When an east-west oriented Mei-Yu front passes over the CMR (which is oriented from north-northeast to south-southwest) from the north, the Mei-Yu front at the surface splits into eastern and
69
western flanks, with an eastern flank that accelerates and a western flank that is retarded.
In addition to the mountain-induced anticyclonic circulation, the retardation of the left flank of the Mei-Yu front is enhanced by the southwesterly monsoon. Recent nonlinear numerical experiments are able to: (a) reproduce the orographically induced high pressure, front splitting, and anticyclonic circulation during the passage of Mei-Yu front over the CMR; (b) show the important contribution of the nonlinear advection, Coriolis force, pressure gradient force, and friction to the local rate of change of horizontam momentum; and (c) indicate that the front bears no dynamical resemblance to either an orographically trapped density current or a Kelvin wave (Sun and Chern 2006).
The dynamics of orographic distortion of fronts can be understood by taking a passive scalar approach, in which an initially straight front comes under the influence of the anticyclonic circulation induced by the mountain, and experiences an anticyclonic turning. This is analogous to the passage of a cyclone over a high mountain, as discussed in Section 5.5. As mentioned earlier, as soon as the front impinges on the mountain, an anticyclonic circulation is produced in the cold postfrontal air and the front advances faster on the left (facing the frontal movement) side of the mountain and is retarded on the right side. Consequently, weakening of the front (frontolysis) is dominant on the left side and lee side facing the direction of frontal propagation, while strengthening of the front (frontogenesis) occurs on the right side. It is found that the degree of blocking on a front along the upwind slope is determined by the Froude number (F = U/Nh) and Rossby number (Ro = U/fa, where a is the half-width or horizontal scale of the mountain) (e.g.,
Davies 1984; Gross 1994).
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5.6.2 Coastally trapped disturbances
A Coastally trapped disturbance (CTD) is a mesoscale disturbance associated with a high pressure surge along a coastal mountain range, confined laterally against the mountain barrier and vertically by stratification, and propagating along the mountain such that the mountain is on the right in the Northern Hemisphere. On the U.S. west coast, a
CTD forms in spring and summer and propagates northward, and appears as a transition from the prevailing northerly flow. The CTD has a length scale of about 1000 km along the shore, 100 km across the shore, and 0.5 km in the vertical. It normally lasts for 2 to 3 days. Figure 5.40 illustrates the formation and evolution of CTD along the U.S. west coast. Before the CTD event, the climatological basic state consists of high pressure in the north-central Pacific and a sloping marine boundary layer accompanied by a northerly jet that has its maximum amplitude at the coast, where the boundary layer intersects the coastal mountains (Fig. 5.40a). At the early stage of the CTD event (Fig. 5.40b), there is eastward movement of the northcentral Pacific high onshore to the Pacific Northwest, possibly accompanied by westward movement of the low in southwest U.S. This synoptic evolution produces offshore flow and a mesoscale coastal low which evacuates the marine layer. As the low-level flow comes into geostrophic balance with the mesoscale low, the westerly flow to the south of the low encounters the coastal mountains and elevates the marine boundary layer. This produces a new mesoscale high to the south of the mesoscale low and pushes the marine boundary layer northward as a trapped disturbance (Fig. 5.40c), producing coastal southerlies as the coastal low becomes elongated and is displaced offshore.
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During its initial or formation stage, as described above, a CTD behaves like a Kelvin wave (Skamarock et al. 1999). A Kelvin wave is a low-frequency inertia-gravity wave trapped in a lateral boundary, such as a mountain barrier in the atmosphere or a continental shelf in the ocean, which propagates counterclockwise in the Northern
Hemisphere around a basin. In the later stage, the CTD is transformed into a density current due to diurnal radiative forcing and differential heating across the coast (Reason et al. 2001). The transition to density current in the later stage might also be affected by other factors, such as nonlinearity and mixing.
A CTD is also known as an orographic jet or ducted coastal ridging in meteorology and is called coastally trapped density current or coastal jet in oceanography. CTDs have also been observed in southeastern Australia, southern Africa, and the southwest coast of
South America (Holland and Leslie 1986). Ahead of the coastal ridging in Australia, there often exists a surge of cold air with squally winds. This phenomenon is called the southerly buster.
5.6.3 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.
Figure 5.41a 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.
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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 being 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.
Figure 5.41b shows a conceptual model of the cold-air damming occurred to the east of the Applachian Mountains at the mature phase. The low-level wind (LLWM in the figure) moved southward at a speed of about 15 ms-1 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.
As described above, 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 (Fig. 5.41b), similar to that shown in Fig. 5.36a 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
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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). It was found that the cold dome shrinks as the Froude number (U/ Nh ) increases or, to a minor degree, as the Ekman number ( /(fh2 ) , where 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.
5.6.4 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 ( F U/ Nh ), 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
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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 (Fig. 5.42; 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).
Appendix 5.1: Some mathematical techniques and relations
(a) Fourier Transform
The Fourier transform of f (x) is defined as
1 fˆ(k) f (x) eikxdx and (A5.1.1a) 2
f (x) fˆ(k) eikxdk . (A5.1.1b)
One of the advantages of the Fourier transform is that it is able to distinguish the up- and down-going waves. An alternative Fourier transform pair may be defined as
and (A5.1.2a)
f (x) 2Re fˆ(k) eikxdk , (A5.1.2b) 0 for real function f. This is also called the one-sided Fourier transform (Queney et al.
1960). Occasionally, the following pair of Fourier transform has been adopted in the literature,
75
1 fˆ(k) f (x) eikxdx and (A5.1.3a)
f (x) Re fˆ(k) eikxdk 0 . (A5.1.3b)
Other variations of Fourier transform pairs, such as using eikx ( eikx) in the forward
(inverse or backward) Fourier transform, have also been used in the literature. No matter which form of the Fourier transform is used, the Fourier transform pair should be able to transform the original function back to itself after performing the forward and inverse transforms.
(b) Jordan’s Lemma
If
lim f (z) 0, R then
lim f (z)eikzdz 0, (A5.1.4) R R for k 0 and where R is an anticlockwise contour of a semicircle above the Re z axis with radius R . Jordan’s Lemma is very useful for converting a line integral to a contour integral.
(c) Riemann-Lebesgue Lemma
lim fˆ(k)eikxdk 0 if fˆ(k) is smooth. (A5.1.5) x
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A smooth function here means that the function is ordinary and absolutely integrable.
The above conclusion is reached by the reasoning of cancellation.
(d) Parseval Theorem
1 f (x)g *(x)dx fˆ(k)gˆ *(k)dk , (A5.1.6) 2 where “^” indicates the Fourier transformed functions. The Parseval theorem is useful for computing wave energy and momentum flux.
References
Aebischer, U. and C. Schar , 1998: Low-level potential vorticity and cyclogenesis to the
lee of the Alps. J. Atmos. Sci., 55, 186-207.
Bacmeister, J. T., and R. T. Pierrehumbert, 1988: On high-drag states of nonlinear
stratified flow over an obstacle. J. Atmos. Sci., 45, 63-80.
Batchelor, G. K., 1967: An Introduction to Fluid Dynamics. Cambridge University Press,
615pp.
Bell, G. D., and L. F. Bosart, 1988: Appalachian cold-air damming. Mon. Wea. Rev.,
116, 137-161.
Blackmon, M., J. M. Wallace, N. C. Lau, and S. Mullen, 1977: An observational study of
the Northern Hemisphere wintertime circulation. J. Atmos. Sci., 34, 1040-1053.
Bleck, R., and C. Mattocks, 1984: A preliminary analysis of the role of potential vorticity
in Alpine lee cyclogenesis. Beitr. Phys. Atmos., 57, 357-368.
Buzzi, A., and S. Tibaldi, 1977: Inertial and frictional effects on rotating stratified flow
over topography. Quart. J. Royal Meteor. Soc., 103, 135-150.
77
Buzzi, A., A. Trevisan, S. Tibaldi, and E. Tosi, 1987: A unified theory of orographic
influences upon cyclogenesis. Meteor. Atmos. Phys., 36, 91-107.
Carlson, T. N., 1998: Mid-Latitude Weather Systems. Amer. Meteor. Soc., 507pp.
Chang, S. W.-J., 1982: The orographic effects induced by an island mountain range on
propagating tropical cyclones. Mon. Wea. Rev., 110, 1255-1270.
Chen, Y.-L., and N. B.-F. Hui, 1992: Analysis of a relatively dry front during the Taiwan
Area Mesoscale Experiment. Mon. Wea. Rev., 120, 2442-2468.
Chung, C. S., K. Hage, and E. Reinelt, 1976: On lee cyclogenesis and airflow in the
Canadian Rocky Mountains and the East Asian Mountains. Mon. Wea. Rev., 104,
878-891.
Clark, T. L., and W. R. Peltier, 1984: Critical level reflection and the resonant growth of
nonlinear mountain waves. J. Atmos. Sci., 41, 3122-3134.
Crook, N. A., T. L. Clark, and M. W. Moncrieff, 1990: The Denver cyclone. Part I:
Generation in low Froude number flow. J. Atmos. Sci., 47, 2725-2742.
Davis, C. A., 1997: The modification of baroclinic waves by the Rocky Mountains. J.
Atmos. Sci., 54, 848-868.
Davies, H. C., 1984: On the orographic retardation of a cold front. Beitr. Phys. Atmos.,
57, 409-418.
Dickinson, M. J., and D. J. Knight, 1999: Frontal interaction with mesoscale topography.
J. Atmos. Sci., 56, 3544-3559.
Doyle, J. D., and D. R. Durran, 2004: Recent developments in the theory of atmospheric
rotors. Bull. Amer. Meteor. Soc., 85, 337-342.
78
Durran, D. R., 1986a: Another look at downslope windstorms. Part I: The development of
analogs to supercritical flow in an infinitely deep, continuously stratified fluid. J.
Atmos. Sci., 43, 2527-2543.
Durran, D. R., 1986b: Mountain waves. In Mesoscale Meteorology and Forecasting, P. S.
Ray (Ed.), Amer. Meteoro. Soc., 472-492.
Durran, D. R., 1990: Mountain waves and downslope winds. Atmospheric Processes
over Complex Terrain, Meteor. Monogr., No. 45, Amer. Meteor. Soc., 59-81.
Durran, D. R., and J. B. Klemp, 1987: Another look at downslope winds. Part II:
Nonlinear amplification beneath wave-overturning layers. J. Atmos. Sci., 44, 3402-
3412.
Eady, E. T., 1949: Long waves and cyclone waves. Tellus, 1, 33-52.
Egger, 1972: Numerical experiments on lee cyclogenesis in the Gulf of Genoa. Beitr.
Phys. Atmos., 45, 320-346.
Epifanio, C. C., 2003: Lee vortices. Encyclopedia of the Atmospheric Sciences,
Cambridge University Press, 1150-1160.
Epifanio, C. C., and D. R. Durran, 2001: Three-dimensional effects in high-drag-state
flows over long ridges. J. Atmos. Sci., 58, 1051-1065.
Epifanio, C. C., and R. Rotunno, 2005: The dynamics of orographic wake formation in
flows with upstream blocking. J. Atmos. Sci., 62, 3127-3150.
Garner, S. T., 1995: Permanent and transient upstream effects in nonlinear stratified flow
over a ridge. J. Atmos. Sci., 52, 227-246.
Gaberšek, S., and D. R. Durran 2004: Gap flows through idealized topography. Part I:
Forcing by large-scale winds in the nonrotating limit. J. Atmos. Sci., 61, 2846-2862.
79
Grimshaw, R. H., and N. Smyth, 1986: Resonant flow of a stratified fluid over
topography. J. Fluid Mech., 169, 429-464.
Gross, B. D., 1994: Frontal interaction with isolated topography. J. Atmos. Sci., 51, 1480-
1496.
Grubišić, V., and J. M. Lewis, 2004: Sierra Wave Project revisited. Bull. Amer. Meteor.
Soc., 85, 1127-1142.
Hayes, J. L., R. T. Williams, and M. A. Rennick, 1993: Lee cyclogenesis. Part II:
Numerical studies. J. Atmos. Sci., 50, 2354-2368.
Haynes, P. H., and M. E. McIntyre, 1990: On the conservation and impermeability
theorems for potential vorticity. J. Atmos. Sci., 47, 2021-2031.
Holland, G. J., and L. M. Leslie, 1986: Ducted coastal ridging over S. E. Australia.
Quart. J. Roy. Meteor. Soc., 112, 731-748.
Hoskins, B. J., M. E. McIntyre, and A. W. Robertson, 1985: On the use and significance
of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc., 111, 877-946.
Hunt, J. C. R, and W. H. Snyder, 1980: Experiments on stably and neutrally stratified
flow over a model three-dimensional hill. J. Fluid Mech., 96, 671-704.
Lee, T.-Y., Y.-Y. Park, and Y.-L. Lin, 1998: A numerical modeling study of mesoscale
cyclogenesis to the east of the Korean Peninsula. Mon. Wea. Rev., 126, 2305-2329.
Lin, Y.-L., and D. J. Perkey, 1989: Numerical modeling studies of a process of lee
cyclogenesis. J. Atmos. Sci., 46, 3685-3697.
Lin, Y.-L., and T.-A. Wang, 1996: Flow regimes and transient dynamics of two-
dimensional stratified flow over an isolated mountain ridge. J. Atmos. Sci., 53, 139-
158.
80
Lin, Y.-L., N.-H. Lin, and R. P. Weglarz, 1992: Numerical modeling studies of lee
mesolows, mesovortices, and mesocyclones with application to the formation of
Taiwan mesolows. Meteor. Atmos. Phys., 49, 43-67.
Lin, Y.-L., J. Han, D. W. Hamilton, and C.-Y. Huang, 1999: Orographic influence on a
drifting cyclone. J. Atmos. Sci., 56, 534-562.
Lin, Y.-L., S.-Y. Chen, C. M. Hill, and C.-Y. Huang, 2005: Control parameters for track
continuity and deflection associated with tropical cyclones over a mesoscale
mountain. J. Atmos. Sci., 62, 1849-1866.
Long, R. R., 1953: Some aspects of the flow of stratified fluids. Part I. A theoretical
investigation. Tellus, 5, 42-58.
Mayr, G.J., 2005: Gap flows - our state of knowledge at the end of MAP. Croatian
Meteor. J., 40 , 6-10.
Miles, J. W., 1968: Lee waves in a stratified flow. Part II: Semi-circular obstacle. J. Fluid
Mech., 33, 803-814.
Miles, J. W., and H. E. Huppert, 1969: Lee waves in a stratified flow. Part 4:
Perturbation approximation. J. Fluid Mech., 35, 497-525.
Orlanski, I., and B. D. Gross, 1994: Orographic modification of cyclone development. J.
Atmos. Sci., 51, 589-611.
Petterssen, S., 1956: Weather Analysis and Forecasting. Vol. I, McGraw Hill, 428pp.
Pichler, H., and R. Steinacker, 1987: On the synoptics and dynamics of orographically
induced cyclones in the Mediterranean. Meteor. Atmos. Phys., 36, 108-117.
Pierrehumbert, R. T., 1986: Lee cyclogenesis. In Mesoscale Meteorology and
Forecasting, P. S. Ray (Ed.), Amer. Meteor. Soc., 493-515.
81
Pierrehumbert, R. T., and B. Wyman, 1985: Upstream effects of mesoscale mountains. J.
Atmos. Sci., 42, 977-1003.
Queney, P., 1948: The problem of air flow over mountain: A summary of theoretical
studies. Bull. Amer. Meteor. Soc., 29, 16-26.
Queney, P., G. Corby, N. Gerbier, H. Koschmieder, and J. Zierep, 1960: The airflow over
mountains. WMO Tech. Note, 34, 135 pp.
Reason, C. J. C., K. J. Tory and P. L. Jackson, 2001: A model investigation of the
dynamics of a coastally trapped disturbance. J. Atmos. Sci., 58, 1892–1906.
Rottman, J. W., and R. B. Smith, 1989: A laboratory model of severe downslope winds.
Tellus, 41A, 401-415.
Schär, C., 1990: Quasi-geostrophic lee cyclogenesis. J. Atmos. Sci., 47, 3044-3066.
Schär, C., 1993: A generation of Bernoulli’s theorem. J. Atmos. Sci., 50, 1437-1443.
Schär, C., and D. R. Durran, 1997: Vortex formation and vortex shedding in continuously
stratified flows past isolated topography. J. Atmos. Sci., 54, 534-554.
Schär, C., and R. B. Smith, 1993a: Shallow-water flow past isolated topography. Part I:
Vorticity production and wake formation. J. Atmos. Sci., 50, 1373-1400.
Schär, C., and R. B. Smith, 1993b: Shallow-water flow past isolated topography. Part II:
Transition to vortex shedding. J. Atmos. Sci., 50, 1401-1412.
Scinocca, J. F., and W. R. Peltier, 1993: The instability of Long’s stationary solution and
the evolution toward severe downslope windstorm flow. Part I: Nested grid numerical
simulations. J. Atmos. Sci., 50, 2245-2263.
Scorer, R. S., 1949: Theory of waves in the lee of mountains. Quart. J. Roy. Meteor.
Soc., 75, 41-56.
82
Scorer, R. S., 1954: Theory of airflow over mountains: III - airstream characteristics.
Quart. J. Roy. Meteor. Soc., 80, 417-428.
Sharman, R. D., and M. G. Wurtele, 1983: Ship waves and lee waves. J. Atmos. Sci., 40,
396-427.
Skamarock, W. C., R. Rotunno, and J. B. Klemp, 1999: Models of coastally trapped
disturbances. J. Atmos. Sci., 56, 3349-3365.
Smith, R. B., 1979: The influence of mountains on the atmosphere. Adv. Geophys., 21,
Ed. B. Saltzman, Academic Press, NY, 87-230.
Smith, R. B., 1980: Linear theory of stratified hydrostatic flow past an isolated mountain.
Tellus, 32, 348-364.
Smith, R. B., 1984: A theory of lee cyclogenesis. J. Atmos. Sci., 41, 1159-1168.
Smith, R. B., 1985: On severe downslope winds. J. Atmos. Sci., 42, 2597-2603.
Smith, R. B., 1986: Further development of a theory of lee cyclogenesis. J. Atmos. Sci.,
43, 1582-1602.
Smith, R. B., 1989a: Hydrostatic airflow over mountains. Adv. Geophys., B. Saltzman
(Ed.), 31 , Academic Press, NY, 1-41.
Smith, R. B., 1989b: Comments on “Low Froude number flow past three-dimensional
obstacles. Part I: Baroclinically generated lee vortices.” J. Atmos. Sci., 46, 3611-3613.
Smolarkiewicz, P. K., and R. Rotunno, 1989: Low Froude number flow past three-
dimensional obstacles. Part I: Baroclinically generated lee vortices. J. Atmos. Sci., 46,
1154-1164.
83
Speranza, A., A. Buzzi, A. Trevisan, and P. Malguzzi, 1985: A theory of deep
cyclogenesis in the lee of the Alps: Part I, Modification of baroclinic instability by
localized topography. J. Atmos. Sci., 42, 1521-1535.
Stein, J., 1992: Investigation of the regime diagram of hydrostatic flow over a mountain
with a primitive equation model. Part I: Two-dimensional flows. Mon. Wea. Rev.,
120, 2962-2976.
Steinacker, R., 1981: Analysis of the temperature and wind fields in the Alpine region. Geophys. Astrophys. Fluid Dyn., 17, 51-62. Sun, W.-Y., and J. D. Chern, 2006: Numerical study of the influence of Central Mountain
Ranges in Taiwan on a cold front. J. Meteor. Soc. Japan, 84, 27-46.
Sun, W.-Y., J. D. Chern, C. C. Wu, and W.-R. Hsu, 1991: Numerical simulation of
mesoscale circulation in Taiwan and surrounding area. Mon. Wea. Rev., 119, 2558-
2573.
Tafferner, A., and J. Egger, 1990: Test of lee cyclogenesis: ALPEX cases. J. Atmos. Sci.,
47, 2417-2428.
Teixeira, M.A.C. and Miranda, P.M.A., 2006: A linear model of gravity wave drag for
hydrostatic sheared flow over elliptical mountains. Quart. J. Roy. Met. Soc., 132,
2439-2458.
Teixeira, M.A.C., Miranda, P.M.A., Argain, J.L. and Valente, M.A., 2005: Resonant
gravity wave drag enhancement in linear stratified flow over mountains, Quart. J.
Roy. Met. Soc., 131, 1795-1814.
Tibaldi, S., A. Buzzi, and A. Speranza, 1990: Orographic cyclogenesis. In Extratropical
Cyclones, C. Newton and E. O. Holopainen (Eds.), Amer. Meteor. Soc., 107-128.
84
Wang, S.-T., 1980: Prediction of the movement and strength of typhoons in Taiwan and
its vicinity. Res. Rep., 108, National Sci. Council Tech. Rep., Taipei, Taiwan, 100pp.
Wang, T.-A., and Y.-L. Lin, 1999: Wave ducting in a stratified shear flow over a two-
dimensional mountain. Part II: Implication for the development of high-drag states
for severe downslope windstorms. J. Atmos. Sci., 56, 437-452.
Xu, Q., S. Gao, and B. H. Fiedler, 1996: A theoretical study of cold air damming with
upstream cold air inflow. J. Atmos. Sci., 53, 312-326.
Zängl, G., 2002: Stratified flow over a mountain with a gap: Linear theory and numerical
simulations. Quart. J. Roy. Meteor. Soc., 128, 927-949.
Problems
5.1 Derive the solutions, (5.1.21) - (5.1.23). Verify the maxima and minima of
different variables denoted in Fig. 5.1a.
5.2 Show that the vertical energy flux, p' w', for terms C in (5.1.28) is positive.
5.3 Prove (5.1.35).
5.4 (a) Based on (5.2.12) and using the Boussinesq approximation, find the solutions of
other variables, w' , u' , p' , and ' , for a hydrostatic mountain wave. (b)
Determine the maxima and minima of , , and near the mountain surface.
5.5 Find the wavelength of the resonance waves of the Scorer's (1949) model with
2 6 -2 2 6 -2 H 2.7 km , lx1 2.12 10 m , and lx2 0.33 10 m , by solving (5.2.35)
graphically for k* using a graphics calculator or spreadsheet program. [Hint:
When plotting the functions with either graphics software or a graphics calculator,
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2 2 2 you must recall from the text that l2 k l1 and restrict your plotting range to
between l1 and l2 .]
5.6 Solve (5.2.34) with the interface conditions to obtain the solution (5.2.39).
5.7 (a) What is the corresponding stratified-flow Froude number (F) and non-
dimensional mountain height (M) for the flow in Figure 5.9a? To find M, you must
estimate H0 from Fig. 5.9a, noting that H = H0. With the values of F and M you
calculate, determine the flow regime from Fig 3.3. (b) Using Fig. 5.15 and based
on Smith’s theory (Eq. 5.3.14), calculate the pressure drag ( D ) on the mountain per
-3 unit length, assuming o 1 kg m and a depth of lower downstream dividing
streamline, H1 3.14 km .
5.8 Show that with vertical wind shear, (5.4.6) becomes
2 2 2 2 l wxx (N /U )wyy 0 , where l is the Scorer parameter [(5.1.6)].
-1 -4 5.9 Prove (5.5.5). Based on (5.5.5) with hm = 500 m, a = 2000 km, N = 0.01 s , f = 10
s-1, make a plot and compare with the sketch of Fig. 5.26.
5.10 (a) Derive (5.5.9) from (5.5.7) and the geostrophic wind and thermal wind relations.
Let ugg U u , vgg V v , U Usz U z , and V Vsz V z . (b) Derive (5.5.10)
by assuming the flow is steady and two-dimensional and taking the Fourier
transform to the system (5.5.8) and (5.5.9).
Figure captions
Fig. 5.1: The steady-state, inviscid flow over a two-dimensional sinusoidal mountain
when (a) lk22 (or N kU ), where k is the terrain wavenumber (= 2/ a , where a
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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 ' (W and C) are also denoted in the figures.
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.
Fig. 5.3: Streamlines of steady state flow over an isolated, bell-shaped mountain when (a)
lk22 (or Na U ), where a is the half-width of the mountain, or (b) lk22 (or
Na U ). (Adapted after Durran 1990)
Fig. 5.4: 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 ( k l, but not k l
-1 -1 ). The flow and orographic parameters are: U = 10 ms , N = 0.01 s , hm = 1 km, and
a = 1 km. (Adapted after Queney 1948)
Fig. 5.5: 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)
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Fig. 5.6: 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)
Fig. 5.7: 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)
Fig. 5.8: 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)
Fig. 5.9: (a) Streamlines for Long’s model solution over a bell-shaped mountain with U =
-1 -1 5 ms , N = 0.01 s , hm = 500 m and a = 3 km; and (b) same as (a) except with a = 1
km. An iterative method is adopted in solving the nonlinear equation (5.3.2) with the
nonlinear lower boundary condition (5.3.3) applied. Note that the dispersive tail of
the nonhydrostatic waves is present in the narrower mountain (case (b)). (Adapted
from Laprise and Peltier 1989a)
Fig. 5.10: Nonlinear flow regimes for a two-dimensional, hydrostatic, uniform flow over
a bell-shaped mountain as simulated by a numerical model, based on the Froude
number ( F U/ Nh ). F varies from 0.5 to 1.3, which gives four different flow
regimes as discussed in the text. Displayed are the fields (left two columns) and
the u'fields (right two columns) for two nondimensional times Ut/ a 12.6 and 50.4.
The dimensional parameters are: N = 0.01 ms-1, h = 1 km, a = 10 km, and U = 5, 7,
11, and 13 ms-1 correspondig to F = 0.5, 0.7, 1.1, and 1.3, respectively. A constant
88
nondimensional physical domain height of 1.7z (where z 2/UN) is used. Both
the abscissa and ordinate in the small panels are labeled in km. (Adapted after Lin
and Wang 1996)
Fig. 5.11: Resonant amplification mechanism for severe downslope winds developed for
basic flow with a prescribed critical level ( zi ) over a bell-shaped mountain.
Displayed are the evolution of the Reynolds stress, ouw'', profile for on-resonance
flows with: ziz/ : (a) 0.75 and (c) 1.75 and for the off-resonance flow with (b)
ziz/ 1.15 , where zo 2/UN. The flow and orographic parameters are:
1 -1 Ns 0.02 , U( z ) Uoi tanh[( z z ) / b ] with Uo 8 ms , b = 600 m,
22 Rimin N/( Uo / b ) 2.25 (minimum Ri), h = 300 m, and a = 3 km. The Froude
number (Uo / Nh ) is 1.33. Height (z) is in km. The profiles in the figure range in time
from 0 to 2880 t , 1440 to 2880 , and 2080 to 4240 for panels (a) to (c),
respectively, where t 5 s. Some of the profiles are sequentially numbered from
earliest to latest (labeled by small numbers). (After Clark and Peltier 1984)
Fig. 5.12: Three distinct stages for the development of severe downslope winds, as
revealed by the triply-nested numerical simulations. Long’s (1953) nonlinear
analytical solution is used to initiate the flow. Displayed are the potential temperature
fields over the lee slope at model times of (a) 0, (b) 20, (c) 66, (d) 96, (e) 160, and (f)
166 min. The grid resolutions for the outer, middle, and inner domains are 500 m, 50
2 m, and 16 3 m, respectively. A bell-shaped mountain with h = 165 m and a = 3 km
is used. The upstream flow parameters are U = 3.3 ms-1 and N = 0.02 s-1. Thus, F =
U/Nh = 1. (Adapted after Scinocca and Peltier 1993)
89
Fig. 5.13: 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)
Fig. 5.14: 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)
Fig. 5.15: 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 ( c ), which is called
dividing streamline. Ho and H1 denote the heights of upstream dividing streamline
and downstream lower dividing streamline, respectively. (b) An example of
transitional flow over a mountain. The dimensional values of the flow and orographic
-1 -1 parameters are U = 20 ms , N = 0.01 s , Ho = 9.42 km, and h = 2 km. This gives
F U/ Nh 1.0. (Adapted after Smith 1985)
Fig. 5.16: The dependence of high-drag states on the lower-layer depth, 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)
90
Fig. 5.17: Three-dimensional, linear, hydrostatic stratified flow over a bell-shaped
mountain (5.4.15) with F U / Nh 100 . The basic flow is from left to right.
Displayed are the nondimensional vertical displacement at ~z Nz /U : (a) / 4 and
(b) . U-shaped disturbances are associated with the upward propagating wave
energy. Solid and dashed curves represent positive and negative values of vertical
displacement. The cross marks the position of the mountain peak. The bold, dashed
circle is the topographic contour at r a , where r is the distance (radius) from the
center of the mountain. These wave patterns are computed by evaluating (5.4.18)
numerically using a two-dimensional FFT. (Adapted after Smith 1980)
Fig. 5.18: Satellite imagery of three-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 sketched in Fig. 5.19. (From Visible Earth,
NASA)
Fig. 5.19: Schematic of transverse (bold-dashed) and diverging (solid) phase lines for a
classical deep water ship wave. (Adapted after Sharman and Wurtele 1983)
Fig. 5.20: (a) Side view of the mean surface shear stress pattern and streamlines on the
center plane of symmetry for a three-dimensional, stratified, viscous flow with
F U / Nh 0.4 past an obstacle with circular contours (e.g., bold solid curve in
(b)). In the figure, N and S denote nodes and saddle points, respectively, and
subscripts a and s denote attachment and separation, respectively. (b) As (a) but for a
plane view of the pattern of surface stress. (Adapted after Hunt and Snyder 1980)
Fig. 5.21: Three-dimensional, stratified, uniform flow with no surface friction over a
bell-shaped mountain simulated by a nonlinear numerical model. Surface
91
streamlines, vertical displacements at Nz/ U /4, and streamlines in the vertical
plane y / a 0 after Ut/9 a are shown in (a), (c), and (e), respectively, for the case
with F 0.66 . The same flow fields but for F 0.22 are shown in the right panels
((b),(d) and (f)). The simulated flow fields have reached quasi-steady state. The flow
-1 -1 -1 and orographic parameters are: U 10ms or 3.3ms , N = 0.01 s , h = 1.5 km, and
a = 10 km, which give F 0.66 or 0.22, respectively. The bell-shaped mountain is
prescribed by (5.4.15). (Adapted after Smolarkiewicz and Rotunno 1989)
Fig. 5.22: Regime diagram for three-dimensional, stratified, uniform, hydrostatic flow
over an isolated mountain. The flow regime is controlled by the horizontal mountain
aspect ratio ( ba/ ) and the nondimensional height or the inverse Froude number (
Nh/ U ), where a and b are the mountain scales in along (x) and perpendicular (y) to
the basic flow directions, respectively. Four classes of phenomena of importance in
this type of flow are: (1) linear mountain waves, (2) wave breaking, (3) flow splitting,
and (4) lee vortices. The circles/ellipses represent the mountain contours. (Adapted
after Smith 1989a and Epifanio 2003)
Fig. 5.23: A schematic diagram showing the generation of leeside vorticity by the vertical
tilting of baroclinically generated horizontal vorticity, based on Smolarkiewicz and
Rotunno (1989). The downward (upward) arrow below the adiabatically-induced
cold (warm) region denotes downward (upward) motion. A negative x -vorticity,
0 , is produced over the right side of the upslope baroclinically by the relatively
cold air (b < 0) along the center line and the relatively warm air to the right (facing
downstream), as indicated by (5.4.29). This negative -vorticity is then swept
92
downstream and produces a positive vertical vorticity, 0 , on the right side of the
lee due to the vertical tilting of the x-vorticity, as implied by (5.4.28).
Fig. 5.24: (a) A conceptual model depicting potential vorticity (PV) generation by
turbulence dissipation at stagnation points associated with wave breaking aloft and
upstream blocking. The symbol “^^^” denotes areas of turbulence generated by wave
breaking or blocking. (Adapted after Smith 1989a); (b) Schematic depiction of the
relationship between PV generation and Bernoulli function on an isentropic surface in
steady-state, stratified flow over the wave breaking region. Thin lines are streamlines
and dark-shaded area over the lee slope denotes a localized region of dissipation due
to wave breaking, a hydraulic jump or blocking. The grey shaded area extending
downstream denotes a reduced Bernoulli function. Open arrows denote the PV flux J
associated with the Bernoulli gradient on the isentropic surface as described by
(5.4.35). (From Schär and Durran 1997)
Fig. 5.25: 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)
Fig. 5.26: A sketch of a quasi-geostrophic, stratified flow over a circular mountain. (a)
The vertical vorticity associated with the deformation of vortex tubes is shown. The
solid lines represent trajectories on the vertical cross section through the center of the
mountain. (b) The streamline pattern near the surface for the flow associated with (a)
as seen from above. (Adapted from Smith 1979 and Buzzi and Tibaldi 1977)
Fig. 5.27: Stratified rotating hydrostatic flow over a two-dimensional ridge with the
parameters: a = 100 km, f = 10-4 s-1, U = 10 ms-1, and N = 0.01 s-1. Equation (5.5.2)
93
multiplied by the non-Boussinesq factor o / (z) is solved with an asymptotic
method. This flow belongs to an intermediate regime between the nonrotating flow
regime (e.g., Fig. 5.3b) and the quasi-geostrophic flow regime (e.g., Fig. 5.26a). The
vertical wavelength of the wave is z 2UN / 6.28km . The bottom figure
depicts the lateral deflection of the streamlines. In the figure, Lf 2 U / f 600 km
near the surface. (Adapted after Queney 1948)
Fig. 5.28: (a) Surface streamlines, and (b) vertical cross section of potential temperature
(horizontal curves) and pressure perturbations at the centerline ( y / a 0 ) for a
uniform westerly flow over idealized topography after a non-dimensional time
Ut / a 40 . The Froude number and the Rossby number are 0.4 and 2.15,
respectively. The dimensional values of the flow and orographic parameters are: U =
10 ms-1, f = 5.8 x 10-5 s-1, a = 40 km, and h = 2.5 km. In estimating the Rossby
number, 2a is used. Panels (c) and (d) are the same as (a) and (b) except with no
rotation. The x and y scales are non-dimensionalized by the mountain half-width in
the x direction. (Adapted after Lin et al. 1999)
Fig. 5.29: Percentage frequency of cyclogenesis during winter from 1899 to 1939 in the
Northern Hemisphere in squares of 100,000 km2. (Adapted after Petterssen 1956)
Fig. 5.30: Two types of lee cyclogenesis over the Alps: (a) Southwesterly (Vorderseiten)
type, and (b) Northwesterly (Überströmungs) type, based on upper-level flow. The
bold, solid lines indicate the upper-level flow and the surface fronts are plotted for
two consecutive times, as denoted by 1 and 2. The shaded areas represent the terrain
higher than 1 km. The mountains on the right hand side are the Alps and those on the
left are the Pyrenees. (Adapted after Pichler and Steinacker 1987)
94
Fig. 5.31: Baroclinic lee wave theory of lee cyclogenesis: example of the steady-state
solution (dashed curve), (5.5.11), and transient solution (solid curves), (5.5.12) for an
initially undisturbed, back-sheared flow over a bell-shaped mountain,
2 2 2 -3 h(x) ha /(x a ) . The flow and orographic parameters are: o 1 kg m ,
-1 4 -1 5 -1 N 0.01 s , f 10 s , h = 3 km, a 2.5x10 m , Uo 20 ms and
H *3 5x10 m . (Adapted after Smith 1984)
Fig. 5.32: Orographic modification of baroclinic instability for Alpine lee cyclogenesis:
An example of cyclogenesis generated by a baroclinic wave in the continuous Eady
model, in a long periodic channel with isolated orography. The mountain has a bi-
2 2 2 2 Gaussian form ( eex// a y b ) and the dimensional values of a and b are 1500 km and
500 km, respectively. The shaded oval denotes the orographic contour of e1 of the
maximum height. (a) Streamfunction of orographic perturbation at z 0
(nondimensional); (b) total streamfunction of the modified baroclinic wave at ;
(c) and (d) are as (a) and (b), respectively, but at the middle level, z 0.5. The basic
zonal wind is added in (d). (Adapted after Speranza et al. 1985)
Fig. 5.33: An example of Rockies lee cyclogenesis. Three stages, as summarized in the
text, are clearly shown in the perturbation geopotential fields on 860 mb at 12-h
intervals from 3/18/12Z (1200 UTC 18 March) to 3/20/12Z 1994. Solid (Dashed)
contours denote positive (negative) values, while bold solid contours denote zero
geopotential perturbations. Darker (Lighter) shading denotes warm (cold) areas at
900-mb ' 4 K ( ' 4 K ). (Adapted after Davis 1997)
Fig. 5.34: Application of a linear theory of orographic modification of baroclinic
instability to the Rockies lee cyclogenesis. The streamfunction of the fastest growing
95
mode in the lower layer of a two-layer channel model on a plane, at three times of
evolution of the baroclinic mode are shown, which also reflects the three stages of
Rockies lee cyclogenesis (see text). The channel length and width are 10000 and
6000 km, respectively. The basic state mean velocity is 17 ms-1 in the upper layer
and 0 ms-1 in the lower layer. The area of the mountain with height above he1 is
shaded, where h is the mountain height (2.5 km). (Adapted after Buzzi et al. 1987)
Fig. 5.35: Tropical cyclones traversing the Central Mountain Range (CMR) of Taiwan
with (a) continuous tracks, and (b) discontinuous tracks. A cyclone track is defined
as discontinuous when the original cyclone (i.e. a low pressure and closed cyclonic
circulation) and a lee cyclone simultaneously co-exist. (Adapted after Wang 1980
and Chang 1982)
Fig. 5.36: Effects of shallow orography on cyclone track deflection through the turning of
(a) the basic flow and (b) outer circulation of the cyclone. The solid-circled L and
dash-circled L denote the parent and new cyclones, respectively. The incoming flow
is geostrophically balanced. H denotes the mountain-induced high pressure. The
lower curve in (a) denotes the trajectory of an upstream air parcel. The cyclone is
steered to the northeast by the basic flow velocity Vg . g is the vertical vorticity
associated with the basic flow. In (b), the dashed curve associated with L denotes the
cyclone track. The new cyclone forms on the lee due to vorticity stretching
associated with the cyclonic circulation that is, in turn, associated with the parent
cyclone. VT is the characteristic tangential wind speed of the parent cyclone, such as
the maximum tangential wind (Vmax). The arrow adjacent to denotes the local
vorticity tendency ( / t ). (Panel (b) is adapted after Carlson 1998)
96
Fig. 5.37: Orographic influence on cyclone track deflection for a drifting cyclone in a
uniform flow past an isolated, finite-length mountain (denoted by the mountain
symbol) on an f-plane: (a) weak blocking, and (b) strong blocking. (Adapted after
Lin et al. 2005)
Fig. 5.38: A conceptual model of a two-dimensional frontal passage over a mesoscale
mountain, which indicates the low-level blocking (denoted by B) of the front along
the windward slope, the development of the lee trough (T1) and secondary trough (T2)
along the lee slope, the separation of the upper-level and lower-level frontal waves,
and the coupling (C) of the upper-level frontal wave with the secondary trough in the
lee of the mountain. (Adapted after Dickinson and Knight 1999)
Fig. 5.39: Examples of three-dimensional frontal passage over mesoscale mountains: (a)
Schematic of 3-h surface isochrones of a cold front passing over the Alps (shaded).
Note that as the front passes over the mountain range it is turned anticyclonically.
(Adapted after Steinacker 1981); and (b) Surface analysis of a Mei-Yu front passing
over the CMR (shaded) of Taiwan at 1800 UTC 13 May 1987. Solid and dashed
curves denote the sea level pressures (mb; variation from 1000 mb) and isotherms
(oC), respectively. (Adapted after Chen and Hui 1992)
Fig. 5.40: Schematic of an idealized coastally trapped disturbance (CTD) formation and
evolution. See text for detailed description. In the figure, “CLIMO” stands for
climatological basic state, while “SYNOPTIC” and “MESO” stand for synoptic and
mesoscale states, respectively. (Adapted after Skamarock et al. 1999)
Fig. 5.41: (a) 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
97
full barb and one pennant representing 5 ms-1 and 25 ms-1, respectively. (b)
Conceptual model of a mature cold-air damming event. LLWM stands for low-level
wind maximum. (Adapted after Bell and Bosart 1988)
Fig. 5.42: Horizontal streamlines and normalized perturbation velocity ()/u U U 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)
98
99
Figure captions
Fig. 5.1: The steady-state, inviscid flow over a two-dimensional sinusoidal mountain
when (a) lk22 (or N kU ), where k is the terrain wavenumber (= 2/ 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 ' (W and C) are also denoted in the figures.
Fig. 5.2: Relations among different mountain wave regimes as determined by l /k , where
l is the Scorer parameter and k is the wave number.
Fig. 5.3: Streamlines of steady state flow over an isolated, bell-shaped mountain when (a)
lk22 (or Na U ), where a is the half-width of the mountain, or (b) lk22 (or
Na U ). (Adapted after Durran 1990)
Fig. 5.4: 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 ( k l, but not k l
-1 -1 ). The flow and orographic parameters are: U = 10 ms , N = 0.01 s , hm = 1 km, and
a = 1 km. (Adapted after Queney 1948)
Fig. 5.5: 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
100
relative to the mountain. (After Smith 1979, reproduced with permission from
Elsvier.)
Fig. 5.6: 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)
Fig. 5.7: 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)
Fig. 5.8: 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)
Fig. 5.9: (a) Streamlines for Long’s model solution over a bell-shaped mountain with U =
-1 -1 5 ms , N = 0.01 s , hm = 500 m and a = 3 km; and (b) same as (a) except with a = 1
km. An iterative method is adopted in solving the nonlinear equation (5.3.2) with the
nonlinear lower boundary condition (5.3.3) applied. Note that the dispersive tail of
the nonhydrostatic waves is present in the narrower mountain (case (b)). (Adapted
from Laprise and Peltier 1989a)
Fig. 5.10: Nonlinear flow regimes for a two-dimensional, hydrostatic, uniform flow over
a bell-shaped mountain as simulated by a numerical model, based on the Froude
number ( F U/ Nh ). F varies from 0.5 to 1.3, which gives four different flow
regimes as discussed in the text. Displayed are the fields (left two columns) and
the u'fields (right two columns) for two nondimensional times Ut/ a 12.6 and 50.4.
101
The dimensional parameters are: N = 0.01 ms-1, h = 1 km, a = 10 km, and U = 5, 7,
11, and 13 ms-1 correspondig to F = 0.5, 0.7, 1.1, and 1.3, respectively. A constant
nondimensional physical domain height of 1.7z (where z 2/UN) is used. Both
the abscissa and ordinate in the small panels are labeled in km. (Adapted after Lin
and Wang 1996)
Fig. 5.11: Resonant amplification mechanism for severe downslope winds developed for
basic flow with a prescribed critical level ( zi ) over a bell-shaped mountain.
Displayed are the evolution of the Reynolds stress, ouw'', profile for on-resonance
flows with: ziz/ : (a) 0.75 and (c) 1.75 and for the off-resonance flow with (b)
ziz/ 1.15 , where zo 2/UN. The flow and orographic parameters are:
1 -1 Ns 0.02 , U( z ) Uoi tanh[( z z ) / b ] with Uo 8 ms , b = 600 m,
22 Rimin N/( Uo / b ) 2.25 (minimum Ri), h = 300 m, and a = 3 km. The Froude
number (Uo / Nh ) is 1.33. Height (z) is in km. The profiles in the figure range in time
from 0 to 2880 t , 1440 to 2880 , and 2080 to 4240 for panels (a) to (c),
respectively, where t 5 s. Some of the profiles are sequentially numbered from
earliest to latest (labeled by small numbers). (After Clark and Peltier 1984)
Fig. 5.12: Three distinct stages for the development of severe downslope winds, as
revealed by the triply-nested numerical simulations. Long’s (1953) nonlinear
analytical solution is used to initiate the flow. Displayed are the potential temperature
fields over the lee slope at model times of (a) 0, (b) 20, (c) 66, (d) 96, (e) 160, and (f)
166 min. The grid resolutions for the outer, middle, and inner domains are 500 m, 50
2 m, and 16 3 m, respectively. A bell-shaped mountain with h = 165 m and a = 3 km
102
is used. The upstream flow parameters are U = 3.3 ms-1 and N = 0.02 s-1. Thus, F =
U/Nh = 1. (Adapted after Scinocca and Peltier 1993)
Fig. 5.13: 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)
Fig. 5.14: 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)
Fig. 5.15: 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 ( c ), which is called
dividing streamline. Ho and H1 denote the heights of upstream dividing streamline
and downstream lower dividing streamline, respectively. (b) An example of
transitional flow over a mountain. The dimensional values of the flow and orographic
-1 -1 parameters are U = 20 ms , N = 0.01 s , Ho = 9.42 km, and h = 2 km. This gives
F U/ Nh 1.0. (Adapted after Smith 1985)
Fig. 5.16: The dependence of high-drag states on the lower-layer depth, 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
103
lower layer resembles: (a) supercritical flow, (b) a propagating hydraulic jump, (c) a
stationary jump, and (d) subcritical flow. (After Durran 1986a)
Fig. 5.17: Three-dimensional, linear, hydrostatic stratified flow over a bell-shaped
mountain (5.4.15) with F U / Nh 100 . The basic flow is from left to right.
Displayed are the nondimensional vertical displacement at ~z Nz /U : (a) / 4 and
(b) . U-shaped disturbances are associated with the upward propagating wave
energy. Solid and dashed curves represent positive and negative values of vertical
displacement. The cross marks the position of the mountain peak. The bold, dashed
circle is the topographic contour at r a , where r is the distance (radius) from the
center of the mountain. These wave patterns are computed by evaluating (5.4.18)
numerically using a two-dimensional FFT. (Adapted after Smith 1980)
Fig. 5.18: Satellite imagery of three-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 sketched in Fig. 5.19. (From Visible Earth,
NASA)
Fig. 5.19: Schematic of transverse (bold-dashed) and diverging (solid) phase lines for a
classical deep water ship wave. (Adapted after Sharman and Wurtele 1983)
Fig. 5.20: (a) Side view of the mean surface shear stress pattern and streamlines on the
center plane of symmetry for a three-dimensional, stratified, viscous flow with
F U / Nh 0.4 past an obstacle with circular contours (e.g., bold solid curve in
(b)). In the figure, N and S denote nodes and saddle points, respectively, and
subscripts a and s denote attachment and separation, respectively. (b) As (a) but for a
plane view of the pattern of surface stress. (Adapted after Hunt and Snyder 1980)
104
Fig. 5.21: Three-dimensional, stratified, uniform flow with no surface friction over a
bell-shaped mountain simulated by a nonlinear numerical model. Surface
streamlines, vertical displacements at Nz/ U /4, and streamlines in the vertical
plane y / a 0 after Ut/9 a are shown in (a), (c), and (e), respectively, for the case
with F 0.66 . The same flow fields but for F 0.22 are shown in the right panels
((b),(d) and (f)). The simulated flow fields have reached quasi-steady state. The flow
-1 -1 -1 and orographic parameters are: U 10ms or 3.3ms , N = 0.01 s , h = 1.5 km, and
a = 10 km, which give F 0.66 or 0.22, respectively. The bell-shaped mountain is
prescribed by (5.4.15). (Adapted after Smolarkiewicz and Rotunno 1989)
Fig. 5.22: Regime diagram for three-dimensional, stratified, uniform, hydrostatic flow
over an isolated mountain. The flow regime is controlled by the horizontal mountain
aspect ratio ( ba/ ) and the nondimensional height or the inverse Froude number (
Nh/ U ), where a and b are the mountain scales in along (x) and perpendicular (y) to
the basic flow directions, respectively. Four classes of phenomena of importance in
this type of flow are: (1) linear mountain waves, (2) wave breaking, (3) flow splitting,
and (4) lee vortices. The circles/ellipses represent the mountain contours. (Adapted
after Smith 1989a and Epifanio 2003)
Fig. 5.23: A schematic diagram showing the generation of leeside vorticity by the vertical
tilting of baroclinically generated horizontal vorticity, based on Smolarkiewicz and
Rotunno (1989). The downward (upward) arrow below the adiabatically-induced
cold (warm) region denotes downward (upward) motion. A negative x -vorticity,
0 , is produced over the right side of the upslope baroclinically by the relatively
cold air (b < 0) along the center line and the relatively warm air to the right (facing
105
downstream), as indicated by (5.4.29). This negative x -vorticity is then swept
downstream and produces a positive vertical vorticity, 0 , on the right side of the
lee due to the vertical tilting of the x-vorticity, as implied by (5.4.28).
Fig. 5.24: (a) A conceptual model depicting potential vorticity (PV) generation by
turbulence dissipation at stagnation points associated with wave breaking aloft and
upstream blocking. The symbol “^^^” denotes areas of turbulence generated by wave
breaking or blocking. (Adapted after Smith 1989a); (b) Schematic depiction of the
relationship between PV generation and Bernoulli function on an isentropic surface in
steady-state, stratified flow over the wave breaking region. Thin lines are streamlines
and dark-shaded area over the lee slope denotes a localized region of dissipation due
to wave breaking, a hydraulic jump or blocking. The grey shaded area extending
downstream denotes a reduced Bernoulli function. Open arrows denote the PV flux J
associated with the Bernoulli gradient on the isentropic surface as described by
(5.4.35). (From Schär and Durran 1997)
Fig. 5.25: 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)
Fig. 5.26: A sketch of a quasi-geostrophic, stratified flow over a circular mountain. (a)
The vertical vorticity associated with the deformation of vortex tubes is shown. The
solid lines represent trajectories on the vertical cross section through the center of the
mountain. (b) The streamline pattern near the surface for the flow associated with (a)
as seen from above. (Adapted from Smith 1979 and Buzzi and Tibaldi 1977)
106
Fig. 5.27: Stratified rotating hydrostatic flow over a two-dimensional ridge with the
parameters: a = 100 km, f = 10-4 s-1, U = 10 ms-1, and N = 0.01 s-1. Equation (5.5.2)
multiplied by the non-Boussinesq factor o / (z) is solved with an asymptotic
method. This flow belongs to an intermediate regime between the nonrotating flow
regime (e.g., Fig. 5.3b) and the quasi-geostrophic flow regime (e.g., Fig. 5.26a). The
vertical wavelength of the wave is z 2UN / 6.28km . The bottom figure
depicts the lateral deflection of the streamlines. In the figure, Lf 2 U / f 600 km
near the surface. (Adapted after Queney 1948)
Fig. 5.28: (a) Surface streamlines, and (b) vertical cross section of potential temperature
(horizontal curves) and pressure perturbations at the centerline ( y / a 0 ) for a
uniform westerly flow over idealized topography after a non-dimensional time
Ut / a 40 . The Froude number and the Rossby number are 0.4 and 2.15,
respectively. The dimensional values of the flow and orographic parameters are: U =
10 ms-1, f = 5.8 x 10-5 s-1, a = 40 km, and h = 2.5 km. In estimating the Rossby
number, 2a is used. Panels (c) and (d) are the same as (a) and (b) except with no
rotation. The x and y scales are non-dimensionalized by the mountain half-width in
the x direction. (Adapted after Lin et al. 1999)
Fig. 5.29: Percentage frequency of cyclogenesis during winter from 1899 to 1939 in the
Northern Hemisphere in squares of 100,000 km2. (Adapted after Petterssen 1956)
Fig. 5.30: Two types of lee cyclogenesis over the Alps: (a) Southwesterly (Vorderseiten)
type, and (b) Northwesterly (Überströmungs) type, based on upper-level flow. The
bold, solid lines indicate the upper-level flow and the surface fronts are plotted for
two consecutive times, as denoted by 1 and 2. The shaded areas represent the terrain
107
higher than 1 km. The mountains on the right hand side are the Alps and those on the
left are the Pyrenees. (Adapted after Pichler and Steinacker 1987)
Fig. 5.31: Baroclinic lee wave theory of lee cyclogenesis: example of the steady-state
solution (dashed curve), (5.5.11), and transient solution (solid curves), (5.5.12) for an
initially undisturbed, back-sheared flow over a bell-shaped mountain,
2 2 2 -3 h(x) ha /(x a ) . The flow and orographic parameters are: o 1 kg m ,
-1 4 -1 5 -1 N 0.01 s , f 10 s , h = 3 km, a 2.5x10 m , Uo 20 ms and
H *3 5x10 m . (Adapted after Smith 1984)
Fig. 5.32: Orographic modification of baroclinic instability for Alpine lee cyclogenesis:
An example of cyclogenesis generated by a baroclinic wave in the continuous Eady
model, in a long periodic channel with isolated orography. The mountain has a bi-
2 2 2 2 Gaussian form ( eex// a y b ) and the dimensional values of a and b are 1500 km and
500 km, respectively. The shaded oval denotes the orographic contour of e1 of the
maximum height. (a) Streamfunction of orographic perturbation at z 0
(nondimensional); (b) total streamfunction of the modified baroclinic wave at ;
(c) and (d) are as (a) and (b), respectively, but at the middle level, z 0.5. The basic
zonal wind is added in (d). (Adapted after Speranza et al. 1985)
Fig. 5.33: An example of Rockies lee cyclogenesis. Three stages, as summarized in the
text, are clearly shown in the perturbation geopotential fields on 860 mb at 12-h
intervals from 3/18/12Z (1200 UTC 18 March) to 3/20/12Z 1994. Solid (Dashed)
contours denote positive (negative) values, while bold solid contours denote zero
geopotential perturbations. Darker (Lighter) shading denotes warm (cold) areas at
900-mb ' 4 K ( ' 4 K ). (Adapted after Davis 1997)
108
Fig. 5.34: Application of a linear theory of orographic modification of baroclinic
instability to the Rockies lee cyclogenesis. The streamfunction of the fastest growing
mode in the lower layer of a two-layer channel model on a plane, at three times of
evolution of the baroclinic mode are shown, which also reflects the three stages of
Rockies lee cyclogenesis (see text). The channel length and width are 10000 and
6000 km, respectively. The basic state mean velocity is 17 ms-1 in the upper layer
and 0 ms-1 in the lower layer. The area of the mountain with height above he1 is
shaded, where h is the mountain height (2.5 km). (Adapted after Buzzi et al. 1987)
Fig. 5.35: Tropical cyclones traversing the Central Mountain Range (CMR) of Taiwan
with (a) continuous tracks, and (b) discontinuous tracks. A cyclone track is defined
as discontinuous when the original cyclone (i.e. a low pressure and closed cyclonic
circulation) and a lee cyclone simultaneously co-exist. (Adapted after Wang 1980
and Chang 1982)
Fig. 5.36: Effects of shallow orography on cyclone track deflection through the turning of
(a) the basic flow and (b) outer circulation of the cyclone. The solid-circled L and
dash-circled L denote the parent and new cyclones, respectively. The incoming flow
is geostrophically balanced. H denotes the mountain-induced high pressure. The
lower curve in (a) denotes the trajectory of an upstream air parcel. The cyclone is
steered to the northeast by the basic flow velocity Vg . g is the vertical vorticity
associated with the basic flow. In (b), the dashed curve associated with L denotes the
cyclone track. The new cyclone forms on the lee due to vorticity stretching
associated with the cyclonic circulation that is, in turn, associated with the parent
cyclone. VT is the characteristic tangential wind speed of the parent cyclone, such as
109
the maximum tangential wind (Vmax). The arrow adjacent to denotes the local
vorticity tendency ( / t ). (Panel (b) is adapted after Carlson 1998)
Fig. 5.37: Orographic influence on cyclone track deflection for a drifting cyclone in a
uniform flow past an isolated, finite-length mountain (denoted by the mountain
symbol) on an f-plane: (a) weak blocking, and (b) strong blocking. (Adapted after
Lin et al. 2005)
Fig. 5.38: A conceptual model of a two-dimensional frontal passage over a mesoscale
mountain, which indicates the low-level blocking (denoted by B) of the front along
the windward slope, the development of the lee trough (T1) and secondary trough (T2)
along the lee slope, the separation of the upper-level and lower-level frontal waves,
and the coupling (C) of the upper-level frontal wave with the secondary trough in the
lee of the mountain. (Adapted after Dickinson and Knight 1999)
Fig. 5.39: Examples of three-dimensional frontal passage over mesoscale mountains: (a)
Schematic of 3-h surface isochrones of a cold front passing over the Alps (shaded).
Note that as the front passes over the mountain range it is turned anticyclonically.
(Adapted after Steinacker 1981); and (b) Surface analysis of a Mei-Yu front passing
over the CMR (shaded) of Taiwan at 1800 UTC 13 May 1987. Solid and dashed
curves denote the sea level pressures (mb; variation from 1000 mb) and isotherms
(oC), respectively. (Adapted after Chen and Hui 1992)
Fig. 5.40: Schematic of an idealized coastally trapped disturbance (CTD) formation and
evolution. See text for detailed description. In the figure, “CLIMO” stands for
climatological basic state, while “SYNOPTIC” and “MESO” stand for synoptic and
mesoscale states, respectively. (Adapted after Skamarock et al. 1999)
110
Fig. 5.41: (a) 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. (b)
Conceptual model of a mature cold-air damming event. LLWM stands for low-level
wind maximum. (Adapted after Bell and Bosart 1988)
Fig. 5.42: Horizontal streamlines and normalized perturbation velocity ()/u U U 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)
Fig. 5.1: The steady-state, inviscid flow over a two-dimensional sinusoidal mountain when (a) lk22 (or N kU ), where k is the terrain wavenumber (= 2/ a , where a is the terrain wavelength), or (b) lk22 (or N kU ). The dashed line in (b) denotes the
111
constant phase line which tilts upstream with height. The maxima and minima of u' , p' (H and L), and ' (W and C) are also denoted in the figures.
112
Fig. 5.2: Relations among different mountain wave regimes as determined by l /k , where l is the Scorer parameter and k is the wave number.
113
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 Fig. 5.3: Streamlines of steady state flow over an isolated, bell-shaped mountain when (a) lk22 (or Na U ), where a is the half-width of the mountain, or (b) lk22 (or Na U ). (Adapted after Durran 1990)
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Fig. 5.4: 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 ( k l, 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)
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Fig. 5.5: 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. (After Smith 1979, reproduced with the permission from Elsvier.)
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Fig. 5.6: 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)
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Fig. 5.7: Lee waves simulated by a nonlinear numerical model for a two-layer airflow over a bell-shaped mountain. Displayed are the quasi-steady state streamlines. In the lower layer (below 5 km approximately), lx27 9 10 m-2, while in the upper layer, lx27 2 10 m-2. (Adapted after Durran 1986b)
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Fig. 5.8: 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, where Nh/ U is the nondimensional mountain height or the reciprocal of the Froude number. Note that the streamlines become vertical in (c) and overturn in (d). (Adapted after Miles 1968)
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Fig. 5.9: (a) Streamlines for Long’s model solution over a bell-shaped mountain with U = -1 -1 5 ms , N = 0.01 s , hm = 500 m and a = 3 km; and (b) same as (a) except with a = 1 km. An iterative method is adopted in solving the nonlinear equation (5.3.2) with the nonlinear lower boundary condition (5.3.3) applied. Note that the dispersive tail of the nonhydrostatic waves is present in the narrower mountain (case (b)). (Adapted from Laprise and Peltier 1989a)
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Fig. 5.10: Nonlinear flow regimes for a two-dimensional, hydrostatic, uniform flow over a bell-shaped mountain as simulated by a numerical model, based on the Froude number ( F U/ Nh ). F varies from 0.5 to 1.3, which gives four different flow regimes as discussed in the text. Displayed are the fields (left two columns) and the u' fields (right two columns) for two nondimensional times Ut/ a 12.6 and 50.4. The dimensional parameters are: N = 0.01 ms-1, h = 1 km, a = 10 km, and U = 5, 7, 11, and 13 ms-1 correspondig to F = 0.5, 0.7, 1.1, and 1.3, respectively. A constant nondimensional physical domain height of 1.7z (where z 2/UN) is used. Both the abscissa and ordinate in the small panels are labeled in km. (Adapted after Lin and Wang 1996)
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Fig. 5.11: Resonant amplification mechanism for severe downslope winds developed for basic flow with a prescribed critical level ( zi ) over a bell-shaped mountain. Displayed are the evolution of the Reynolds stress, ouw'', profile for on-resonance flows with: ziz/ : (a) 0.75 and (c) 1.75 and for the off-resonance flow with (b) ziz/ 1.15 , 1 where zo 2/UN. The flow and orographic parameters are: Ns 0.02 , -1 22 U( z ) Uoi tanh[( z z ) / b ] with Uo 8 ms , b = 600 m, Rimin N/( Uo / b ) 2.25
(minimum Ri), h = 300 m, and a = 3 km. The Froude number (Uo / Nh ) is 1.33. Height (z) is in km. The profiles in the figure range in time from 0 to 2880 t , 1440 to 2880 , and 2080 to 4240 for panels (a) to (c), respectively, where t 5 s. Some of the profiles are sequentially numbered from earliest to latest (labeled by small numbers). (After Clark and Peltier 1984)
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Fig. 5.12: Three distinct stages for the development of severe downslope winds, as revealed by the triply-nested numerical simulations (see text for details). Long’s (1953) nonlinear analytical solution is used to initiate the flow. Displayed are the potential temperature fields over the lee slope at model times of (a) 0, (b) 20, (c) 66, (d) 96, (e) 160, and (f) 166 min. The grid resolutions for the outer, middle, and inner domains are 2 500 m, 50 m, and 16 3 m, respectively. A bell-shaped mountain with h = 165 m and a = 3 km is used. The upstream flow parameters are U = 3.3 ms-1 and N = 0.02 s-1. Thus, F = U/Nh = 1. (Adapted after Scinocca and Peltier 1993)
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Fig. 5.13: 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)
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Fig. 5.14: 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 ; (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)
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Fig. 5.15: 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 ( c ), which is called dividing streamline. Ho and H1 denote the heights of upstream dividing streamline and downstream lower dividing streamline, respectively. (b) An example of transitional flow over a mountain. The dimensional values of the flow and orographic parameters are U = -1 -1 20 ms , N = 0.01 s , Ho = 9.42 km, and h = 2 km. This gives F U/ Nh 1.0 . (Adapted after Smith 1985)
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Fig. 5.16: The dependence of high-drag states on the lower-layer depth, 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)
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Fig. 5.17: Three-dimensional, linear, hydrostatic stratified flow over a bell-shaped mountain (5.4.15) with F U / Nh 100. The basic flow is from left to right. Displayed are the nondimensional vertical displacement at ~z Nz /U : (a) / 4 and (b) . U- shaped disturbances are associated with the upward propagating wave energy. Solid and dashed curves represent positive and negative values of vertical displacement. The cross marks the position of the mountain peak. The bold, dashed circle is the topographic contour at r a , where r is the distance (radius) from the center of the mountain. These wave patterns are computed by evaluating (5.4.18) numerically using a two-dimensional FFT. (Adapted after Smith 1980)
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Fig. 5.18: Satellite imagery of three-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 sketched in Fig. 5.19. (From Visible Earth, NASA)
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Fig. 5.19: Schematic of transverse (bold-dashed) and diverging (solid) phase lines for a deep water ship wave. (Adapted after Sharman and Wurtele 1983)
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Fig. 5.20: (a) Side view of the mean surface shear stress pattern and streamlines on the center plane of symmetry for a three-dimensional, stratified, viscous flow with F U / Nh 0.4 past an obstacle with circular contours (e.g., bold solid curve in (b)). In the figure, N and S denote nodes and saddle points, respectively, and subscripts a and s denote attachment and separation, respectively. (b) As (a) but for a plane view of the pattern of surface stress. (Adapted after Hunt and Snyder 1980)
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Fig. 5.21: Three-dimensional, stratified, uniform flow with no surface friction over a bell-shaped mountain simulated by a nonlinear numerical model. Surface streamlines, vertical displacements at Nz/ U /4, and streamlines in the vertical plane ya/0 after Ut/9 a are shown in (a), (c), and (e), respectively, for the case with F 0.66 . The same flow fields but for F 0.22 are shown in the right panels ((b),(d) and (f)). The simulated flow fields have reached quasi-steady state. The flow and orographic -1 -1 -1 parameters are: U 10ms or 3.3ms , N = 0.01 s , h = 1.5 km, and a = 10 km, which give F 0.66 or 0.22, respectively. The bell-shaped mountain is prescribed by (5.4.15). (Adapted after Smolarkiewicz and Rotunno 1989)
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Fig. 5.22: Regime diagram for three-dimensional, stratified, uniform, hydrostatic flow over an isolated mountain. The flow regime is controlled by the horizontal mountain aspect ratio ( ba/ ) and the nondimensional height or the inverse Froude number ( Nh/ U ), where a and b are the mountain scales in along (x) and perpendicular (y) to the basic flow directions, respectively. Four classes of phenomena of importance in this type of flow are: (1) linear mountain waves, (2) wave breaking, (3) flow splitting, and (4) lee vortices. The circles/ellipses represent the mountain contours. (Adapted after Smith 1989a and Epifanio 2003)
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0 (b < 0) 0 Cold z Warm Warm (b > 0) > 0 < 0 (b > 0) y x U
Fig. 5.23 Fig. 5.23: A schematic diagram showing the generation of leeside vorticity by the vertical tilting of baroclinically generated horizontal vorticity (Smolarkiewicz and Rotunno 1989). The downward (upward) arrow below the adiabatically-induced cold (warm) region denotes downward (upward) motion. A negative x -vorticity, 0 , is produced over the right side of the upslope baroclinically by the relatively cold air (b < 0) along the center line and the relatively warm air to the right (facing downstream), as indicated by (5.4.29). This negative -vorticity is then swept downstream and produces a positive vertical vorticity, 0 , on the right side of the lee due to the vertical tilting of the x- vorticity, as implied by (5.4.28).
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Fig. 5.24: (a) A conceptual model depicting potential vorticity (PV) generation by turbulence dissipation at stagnation points associated with wave breaking aloft and upstream blocking. The symbol “^^^” denotes areas of turbulence generated by wave breaking or blocking. (Adapted after Smith 1989a); (b) Schematic depiction of the relationship between PV generation and Bernoulli function on an isentropic surface in steady-state, stratified flow over the wave breaking region. Thin lines are streamlines and dark-shaded area over the lee slope denotes a localized region of dissipation due to wave breaking, a hydraulic jump or blocking. The grey shaded area extending downstream denotes a reduced Bernoulli function. Open arrows denote the PV flux J associated with the Bernoulli gradient on the isentropic surface as described by (5.4.35). (From Schär and Durran 1997)
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Fig. 5.25: 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)
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Fig. 5.26: A sketch of a quasi-geostrophic, stratified flow over a circular mountain. (a) The vertical vorticity associated with the deformation of vortex tubes is shown. The solid lines represent trajectories on the vertical cross section through the center of the mountain. (b) The streamline pattern near the surface for the flow associated with (a) as seen from above. (Adapted from Smith 1979 and Buzzi and Tibaldi 1977)
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Fig. 5.27: Stratified rotating hydrostatic flow over a two-dimensional ridge with the parameters: a 100km , f 104 s1 , U 10ms 1 , and Ns 0.01 1 . Equation (5.5.2) multiplied by the non-Boussinesq factor o / (z) is solved with an asymptotic method. This flow belongs to an intermediate regime between the nonrotating flow regime (e.g., Fig. 5.3b) and the quasi-geostrophic flow regime (e.g., Fig. 5.26a). The vertical wavelength of the wave is z 2UN / 6.28km . The bottom figure depicts the lateral deflection of the streamlines. In the figure, Lf 2 U / f 600 km near the surface. (Adapted after Queney 1948)
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Fig. 5.28: (a) Surface streamlines, and (b) vertical cross section of potential temperature (horizontal curves) and pressure perturbations at the centerline ( y / a 0 ) for a uniform westerly flow over idealized topography after a non-dimensional time Ut / a 40 . The Froude number and the Rossby number are 0.4 and 2.15, respectively. The dimensional values of the flow and orographic parameters are: U = 10 ms-1, f = 5.8 x 10-5 s-1, a = 40 km, and h = 2.5 km. In estimating the Rossby number, 2a is used. Panels (c) and (d) are the same as (a) and (b) except with no rotation. The x and y scales are non- dimensionalized by the mountain half-width in the x direction. (Adapted after Lin et al. 1999)
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Fig. 5.29: Percentage frequency of cyclogenesis during winter from 1899 to 1939 in the Northern Hemisphere in squares of 100,000 km2. (Adapted after Petterssen 1956)
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Fig. 5.30: Two types of lee cyclogenesis over the Alps: (a) Southwesterly (Vorderseiten) type, and (b) Northwesterly (Überströmungs) type, based on upper-level flow. The bold, solid lines indicate the upper-level flow and the surface fronts are plotted for two consecutive times, as denoted by 1 and 2. The shaded areas represent the terrain higher than 1 km. The mountains on the right hand side are the Alps and those on the left are the Pyrenees. (Adapted after Pichler and Steinacker 1987)
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Fig. 5.31: Baroclinic lee wave theory of lee cyclogenesis: an example of the steady-state solution (dashed curve), (5.5.11), and transient solution (solid curves), (5.5.12) for an initially undisturbed, back-sheared flow over a bell-shaped mountain, 2 2 2 -3 h(x) ha /(x a ) . The flow and orographic parameters are: o 1 kg m , -1 4 -1 5 -1 N 0.01 s , f 10 s , h = 3 km, a 2.5x10 m , Us 20 ms and H *3 5x10 m . (Adapted after Smith 1984)
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Fig. 5.32: Orographic modification of baroclinic instability for Alpine lee cyclogenesis: An example of cyclogenesis generated by a baroclinic wave in the continuous Eady model, in a long periodic channel with isolated orography. The mountain has a bi- 2 2 2 2 Gaussian form ( eex// a y b ) and the dimensional values of a and b are 1500 km and 500 km, respectively. The shaded oval denotes the orographic contour of e1 of the maximum height. (a) Streamfunction of orographic perturbation at z 0 (nondimensional); (b) total streamfunction of the modified baroclinic wave at ; (c) and (d) are as (a) and (b), respectively, but at the middle level, z 0.5. The basic zonal wind is added in (d). (Adapted after Speranza et al. 1985)
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Fig. 5.33: An example of Rockies lee cyclogenesis. Three stages, as summarized in the text, are clearly shown in the perturbation geopotential fields on 860 mb at 12-h intervals from 3/18/12Z (1200 UTC 18 March) to 3/20/12Z 1994. Solid (Dashed) contours denote positive (negative) values, while bold solid contours denote zero geopotential perturbations. Darker (Lighter) shading denotes warm (cold) areas at 900-mb ' 4 K ( ' 4 K ). (Adapted after Davis 1997)
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Fig. 5.34: Application of a linear theory of orographic modification of baroclinic instability to the Rockies lee cyclogenesis. The streamfunction of the fastest growing mode in the lower layer of a two-layer channel model on a plane, at three times of evolution of the baroclinic mode are shown, which also reflects the three stages of Rockies lee cyclogenesis (see text). The channel length and width are 10000 and 6000 km, respectively. The basic state mean velocity is 17 ms-1 in the upper layer and 0 ms-1 in the lower layer. The area of the mountain with height above he1 is shaded, where h is the mountain height (2.5 km). (Adapted after Buzzi et al. 1987)
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Fig. 5.35: Tropical cyclones traversing the Central Mountain Range (CMR) of Taiwan with (a) continuous tracks, and (b) discontinuous tracks. A cyclone track is defined as discontinuous when the original cyclone (i.e. a low pressure and closed cyclonic circulation) and a lee cyclone simultaneously co-exist. (Adapted after Wang 1980 and Chang 1982)
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Fig. 5.36: Effects of shallow orography on cyclone track deflection through the turning of (a) the basic flow and (b) outer circulation of the cyclone. The solid-circled L and dash- circled L denote the parent and new cyclones, respectively. The incoming flow is geostrophically balanced. H denotes the mountain-induced high pressure. The lower curve in (a) denotes the trajectory of an upstream air parcel. The cyclone is steered to the northeast by the basic flow velocity Vg . g is the vertical vorticity associated with the basic flow. In (b), the dashed curve associated with L denotes the cyclone track. The new cyclone forms on the lee due to vorticity stretching associated with the cyclonic circulation that is, in turn, associated with the parent cyclone. VT is the characteristic tangential wind speed of the parent cyclone, such as the maximum tangential wind (Vmax). The arrow adjacent to denotes the local vorticity tendency ( / t ). (Panel (b) is adapted after Carlson 1998)
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Fig. 5.37: Orographic influence on cyclone track deflection for a drifting cyclone in a uniform flow past an isolated, finite-length mountain (denoted by the mountain symbol) on an f-plane: (a) weak blocking, and (b) strong blocking. (Adapted after Lin et al. 2005)
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Fig. 5.38: A conceptual model of a two-dimensional frontal passage over a mesoscale mountain, which indicates the low-level blocking (denoted by B) of the front along the windward slope, the development of the lee trough (T1) and secondary trough (T2) along the lee slope, the separation of the upper-level and lower-level frontal waves, and the coupling (C) of the upper-level frontal wave with the secondary trough in the lee of the mountain. (Adapted after Dickinson and Knight 1999)
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Fig. 5.39: Examples of three-dimensional frontal passage over mesoscale mountains: (a) Schematic of 3-h surface isochrones of a cold front passing over the Alps (shaded). Note that as the front passes over the mountain range it is turned anticyclonically. (Adapted after Steinacker 1981); and (b) Surface analysis of a Mei-Yu front passing over the CMR (shaded) of Taiwan at 1800 UTC 13 May 1987. Solid and dashed curves denote the sea level pressures (mb; variation from 1000 mb) and isotherms (oC), respectively. (Adapted after Chen and Hui 1992)
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(a) t = to (b) t = t 1 H SYNOPTIC
H L H L L CLIMO CLIMO CLIMO MESO CLIMO L SYNOPTIC
H L L L CLIMO CLIMO CLIMOH MESO MESO L CLIMO
(c) t = t 2 H L L L CLIMO MESO MESO CLIMO H MESO L CLIMO L CLIMO MESO CLIMOH L H CLIMO MESO L H MESO
Fig. 5.40: Schematic of an idealized coastally trapped disturbance (CTD) formation and evolution. See text for detailed description. In the figure, “CLIMO” stands for climatological basic state, while “SYNOPTIC” and “MESO” stand for synoptic and mesoscale states, respectively. (Adapted after Skamarock et al. 1999)
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Fig. 5.41: (a) 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. (b) Conceptual model of a mature cold-air damming event. LLWM stands for low-level wind maximum. (Adapted after Bell and Bosart 1988)
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200 (a) (b)
100
0 y (km) y
-100
-200 200200 (c) (d)
100100
00 y y (km)
-100-100
-200-200 -200 -100 0 100 200-200 -100 0 100 200 x (km) x (km)
Fig. 5.42: Horizontal streamlines and normalized perturbation velocity ()/u U U 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)
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