Generation of coastal lows by synoptic-scale waves

Quart. J . R . Met. SOC. (1981), 107, pp. 521-530 551.515.11: 551.588.2 (68)

Generation of coastal lows by synoptic-scale waves

By NGUYEN NGOC ANH and A. E. GILL Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9E W

(Received 12 June 1980; revised 30 September 1980)

SUMMARY

A model is presented in which eastward propagating synoptic waves interact with a rneridional escarpment. It is shown that, under suitable conditions, trapped waves are produced at the escarpment, the dominant wave being poleward propagating. This model is intended to represent the generation of coastally trapped disturbances like those found in southern Africa.

1. INTRODUCTION

One of the dominant features of the weather around the coast of southern Africa is the appearance of low-pressure disturbances called coastal lows which are confined to the coastal area and propagate along the coast at velocities of order 1Oms-'. They are found at all times of year and appear at a given station about once every six days. Various statistical properties of the lows have been calculated by Preston-Whyte and Tyson (1973). Distur- bance intensity is a maximum near the coast and decays seawards. The feature which allows this trapping of energy in the horizontal is the topography, for most of southern Africa lies above the 1 km level and is fringed by a steep escarpment. Energy can also be trapped in the vertical because a strong inversion is always found at a height about 1 km above sea level. Statistics of this inversion are given by Preston-Whyte, Diab and Tyson (1977).

Gill (1977) has suggested that the coastal lows are essentially Kelvin waves generated by interaction of synoptic-scale systems with the topography. Studies of topographically trapped waves with related structure have been made with oceanographic applications in mind. For instance, Longuet-Higgins (1968) considered the trapping of waves along a discontinuity in depth in a homogeneous uniformly rotating ocean. A review has been presented by LeBlond and Mysak (1977).

The idea behind the present study is to construct a simple model in which synoptic- scale systems could interact with a topographic feature similar to the escarpment of southern Africa. The escarpment is represented by a step, as shown in Fig. 1, and the temperature structure is represented by two homogeneous layers as shown. The depth of the lower layer is slightly less than the height of the step, so only one layer is found over the continent.

For convenience, the step is chosen to run north-south and to coincide with the y-axis. The continent occupies the region x > 0. In the undisturbed state, the lower layer, which has density p2, is of depth H2 and at rest. The upper layer has depth H , , density p1 and is presumed to move at a uniform velocity U , towards the east. This ensures that eastward travelling waves (representing synoptic-scale disturbances) are possible away from the step, and it is the interaction of these waves with the step which will be examined.

2. EQUATIONS

The equations used are the standard quasi-geostrophic equations on a beta-plane, and may be derived as follows. Suppose p i (i = 1 for upper layer, i = 2 for lower layer) is the perturbation pressure divided by density, (ui, ui) is the perturbation velocity, and h is the

521

522 NGUYEN NGOC ANH and A. E. GILL

pressure density ? u,

Figure 1 . The two-layer model and coordinate system.

perturbation displacement (positive upwards) of the interface. The fluids will be treated as incompressible and the Boussinesq approximation made. The acceleration due to gravity is g and I is time. The perturbation momentum equations are therefore

Dt-fui Diui = -ax aPi

where f is the Coriolis parameter, and

(2) D,=- a a D 2 - a D t - at ax' ~t a t

+ u - - - - .

It will be assumed that D J D t 6 f, so the momentum equations can, following Brunt and Douglas (1928), be inverted to give velocity in terms of pressure as follows:

The first term is the geostrophic wind and the second term the isallobaric wind. Buoyancy effects only occur because of vertical motion, and this only occurs when there

is convergence or divergence. When the divergence is calculated from Eq. (3) and the beta-plane approximation made, the result is

aui do. p ap , 1 D, a2pi a2pi -+'= -1--1- -+- ax a y j - ax f D t ( a x 2 a y ) (4)

where p = df/dy has the usual meaning. The remaining equations are the hydrostatic equation, which yields

P 2 - P 1 = g ' k . ( 5 )

GENERATION OF COASTAL LOWS BY SYNOPTIC-SCALE WAVES 523

where g' is the reduced gravity given by

g' = ( P 2 - P A S I P 2 ( 6 ) and the continuity equations. The rigid lid approximation is made so that the upper layer equation is

The potential vorticity equation is obtained by substituting the expression Eq. (4) for the divergence in Eq. (7). In addition, it is assumed that the northward flow is, to a first approximation, geostrophic, so one can replace u i byf- ' ap, /ax . Also, the interface slope is related to the current by the thermal-wind equation, i.e.

The result of the substitutions is

Note that although the velocity component u i has been assumed to be approximately geostrophic, the same assumption has nor been made about ui. Thus Eq. (9) has wider validity than assumed under the normal quasi-geostrophic derivation, and this will be exploited later.

In Eq. (9) the first term represents the rate of change of perturbation potential vorticity following the motion. The remaining term represents advection of mean potential vorticity by the perturbation. In the spirit of the beta-plane approximation, the coefficients HI, /3 and f a r e treated as constants in these equations, since they are not presumed to change significantly over the wave scale. The lower-layer equation, obtained similarly, is

The next step is to make the equations non-dimensional by choosing a length scale I , equal to the meridional scale of the waves, a time scale (BL)-' and the corresponding velocity scale pL2. The parameters of the model are therefore

the scaled velocity,

the Rossby radius relative to the given scale, and

6 = H 2 / H l . (13)

the depth ratio of the two layers. Since the lower layer is only about 1 km deep, 6 is quite small (about 1/10). For a meridional wavelength of 3000km, the length scale is about 500 km, the time scale about one day and the velocity scale about 5 m s-'. The value of E is about 0.6, corresponding to a Rossby radius of 300 km.

In addition, wavelike disturbances proportional to

exp(ikx & i y - iot)

524 NGUYEN NGOC ANH and A . E. GILL

are assumed, so Eq. (9) becomes, in non-dimensional form

6(u - C)g’h = {(u - C)E2K2 - E’ -6u}p, (14)

where c = o / k . (15)

and K’ = l + k ‘ . (16)

cg’k = ( I / - E’ - E 2 K ’ C ) P 2 . (17)

Similarly, Eq. (10) becomes

Using Eq. (5) to substitute for p 2 , this becomes

{ C ( l + E 2 K 2 ) + E Z - I / } g ’ h = ( u - E 2 - & ’ K ’ C ) p I . . (18)

Finally, eliminating pI and k between this equation and Eq. (14) the resulting dispersion equation is

{( u - C ) K ’ - l } { C ( l + E 2 K 2 ) + E’ - u> = 6C( 1 + K’C.) (19)

There is an additional equation required for the single layer over the continent. This is the same as Eq. (7) except that h = 0 and dH, /d j , = 0, i.e. the motion is non-divergent. The corresponding form of Eq. (14) is the usual Rossby-wave equation

{(u-C)K2-l}pl = 0. (20)

3. DISPERSION PROPERTIES

Since 6 is small, the right-hand side of the dispersion relation Eq. (19) can be neglected to a first approximation, so it factorizes. The first factor being zero corresponds to a barotropic planetary wave, the dispersion relation being the same as Eq. (20) the one which applies over the continent. The second factor being zero corresponds to a baroclinic wave with dispersion relation

c = o / k = ( U - E ’ ) / ( ~ +E2+E2k2) (21)

For values of 6 - 0.1, this approximation is quite satisfactory, and will be adopted from here onwards. (The correction to the above values of c is of order 6 except where the speeds of the barotropic and baroclinic waves almost coincide. In that case, the correction becomes of order 6* and complex, so baroclinic instability is possible, as discussed, for instance, in Gill, Green and Simmons (1974).)

The main interest centres on the nature of the baroclinic wave which can be generated by interaction of an incident barotropic wave on the step. The frequency o of the baroclinic wave must be equal to that of the incident wave, so Eq. (20) determines the value of k. The equation for k is quadratic, and the solution is

2~’k = ( U - ~ ~ ) / o + i [ 4 ~ ~ ( 1 + E ~ ) - { ( U - E ~ ) / O } ~ ] ~ . (22)

(23)

This shows that trapped waves are possible, provided

I U - E 2 I < 2JOI&( 1 + 2)f, i.e. provided U is less than a critical value U,. In dimensional terms, the expression for U, is thus

U I c = 2wa(1+/2a’)f+pa’, . (24)


where

a ' = g'H,/f' . ( 2 5 )

is the square of the Rossby radius. For typical periods of 4-6 days, U 1 , is between 11 and 15 m s- '. For velocities well below critical, the wave trapping is quite strong. For instance, for a 5-day period, the non-dimensional value of w is about 1.5. If U is 1.6 and E* = 0.4, Eq. (22) gives

k = 1-1.6; . (26)

the minus sign being chosen to give decay away from the coast. The trapping scale is 300 km, i.e. about the same as the Rossby radius.

4. INTERACTlON WlTH THE STEP

To calculate the response to a wave incident on the step, it is necessary to apply appropriate boundary conditions at x = 0. The condition for the lower layer is the vanish- ing of the velocity normal to the boundary while for the upper layer, there are three conditions, namely continuity of pressure, normal velocity and potential vorticity. For small 6, however, the calculation is greatly simplified. Equation (18) shows that for the barotropic wave, p , and g'k are the same order, so Eq. (14) for the barotropic mode becomes an equation for p , only, and the same equation as Eq. (20), the one valid over the continent. It follows that the barotropic wave propagates over the step without change of form.

The three conditions for the upper layer are satisfied by this solution, but the lower- layer condition is not. To satisfy this condition, a baroclinic wave solution must be added. From Eq. (14) p 1 is of order 6 compared with g ' h for such a wave, so the correction due to the presence of a baroclinic wave will not affect the upper layer to a first approximation. The amplitude of the baroclinic wave is, therefore, entirely determined by the boundary condition for the lower layer.

Suppose now the incident synoptic wave has wavenumber (ks, I ) where I = -t 1 and has unit amplitude at the ground. Then the solution has the form

p 2 = exp(i/y - ior){exp(ik,x)+ Aexp(ik,x)} . (27) where ( k , , l ) is the wavenumber of the baroclinic trapped wave. The value of A is determined from the condition u2 = 0 at x = 0, i.e. putting the second of Eq. (3) in non-dimensional form

where

Y = - P U f . (29) is positive in the southern hemisphere (value about 0.15). It follows that A is given by

When k , is expressed in terms of its real part k, , and imaginary part kri, the denominator becomes

I - + iyok,,. (31)


Since k r i is negative, the denominator has smaller magnitude for a southward propagating wave ( I < 0) than for a northward propagating one, so the response is larger. This is because Kelvin waves propagate southwards and a larger response is induced when the incident wave propagates in the same direction. Thus when two incident waves with 1 = 1 and equal amplitudes are added to give an eastward propagating wave with amplitude varying sinusoidally with latitude, the response at the coast is still dominated by a southward propagating trapped wave. For instance, for the case w = 1.5, e2 = 0.4, U = 1.6, y = 0.15 and k, = 1.24, the value of k , is given by k,, = 1 and kr i = - 1.6, and so A is -0.75exp (0.11 i ) for I = I and A = - 1.52exp (0.06i) for I = - 1. Thus the southward trapped wave is twice as big as the northward one. Summing the waves given by Eq. (27) and taking the real part gives for the surface pressure

p2 = ~ C O S Y C O S ( ~ . ~ ~ X - 1.5 t ) -

-0~75exp(l~6x)cos(y+x- 1.5 t+0.11) - (32) -1~52exp(l~6x)cos(x-y- 1.5tt-0.06).

Contours of p 2 are shown in Fig. 2 at intervals of f period. The solution for the interface height can be obtained by calculating the ratio of g’h/p,

for each wave component. The value is unity for the baroclinic wave and can be calculated from Eq. (17) for the barotropic wave. For the example chosen, the ratio in the latter case is small (- 0.02) so the trapped wave is much more prominent in this field. The result is

g ’ h = -0*04cosycos(1~24x-1*5t)-

-0.75 exp( 1.6 x) cos(y + x - 1.5 t + 0.1 1) - (33) - 1.52 exp(l.6 x) cos(x - y - 1.5 t + 0.06)

and this field is also shown in Fig. 2.

6

5

4

3

2

1

0 -6 -5 - 4 -3 -2 -1 0

Fig. 2. (See facing page.) -3 -2 -1 0


S - -

-

4 -

3 -

2-

- -

-6 -5 - 4 -3 -2 -1 0 -3 -2 -1 0

(C)

3

2 3 -6 -5 -4 -3 -2 -1 0 -3 -2 -1 0

Figure 2. The surface pressure field p 2 (left) given by Eq. (32) and the field g’h (right) given by Eq. (33) where h is the interface height. The panel (a) is at f = 0 and (b) and (c) at intervals of Q period. After the third panel, the pattern of the first panel is repeated with signs reversed. The incident wave field is eastward propagating, but coastal disturbances tend to propagate polewards (to the south). The contour interval

is 0.5 in each case.


For the linear case illustrated in Fig. 2, coastal lows are equally as prominent as coastal highs. The lows do not propagate at uniform speed along the coast, but are quite strongly modulated because of the presence of the northward propagating wave. The influence of the latter wave would perhaps be weaker if the coast were slanted more towards a NW-SE orientation. In the case chosen, the coastal low is not in fact very prominent in the surface pressure pattern, although it stands out very clearly in the interface height pattern. It does, however, illustrate how a southward propagating coastal disturbance can be established when the incident disturbance and the geometry above give no preference to the north or south. Thus the preference for southward propagation comes purely from the dynamics of the trapped wave.

For any given solution, the errors involved in making approximations can be calculated a posteriori. For instance, the error in neglecting the second term compared with the first in the f i rs t of Eq. (3) is given by the ratio y(w/k,- Or) z 0.06 for the barotropic wave (whose properties are determined by the upper layer equation), and by yw/lktl x 0.12 for the trapped wave (for which the motion of the upper layer is neglected because of the small-li approximation). By contrast, the ratio of the second term to the first in the second of Eq. (3) is of order yolk,l z 0.42 for the trapped wave. This ratio is not assumed small. As a conse- quence, the condition u2 = 0 leads to Eq. (28) and the coefficient A given by Eq. (30) is significantly different from unity.

A more extreme case which illuminates the nature of the approximations is the asymp- totic limit obtained when y + 0, e / y is of order unity and U/y + 0. Then all the above approximations are formally valid in the limit as y + 0 and both terms of Eq. (28) are of the same order.

5. ALTERNATIVE MODELS OF THE INCIDENT SYNOPTIC WAVE

The layer beneath the low-level inversion is much thinner than the troposphere, with the result (formalized in the small4 approximation) that synoptic-scale waves which occupy the whole depth of the troposphere are little influenced by the presence or absence of this layer. In the interests of simplicity, the model discussed above included only one layer of tropospheric thickness, so leaving no alternative but to model the incident wave as a barotropic wave.

In practice, the incident synoptic wave would be expected to have a baroclinic structure, so rendering the above model deficient. However, the small4 approximation also implies that the trapped wave is not sensitive to the vertical structure of the incident wave because the energy of the trapped wave is confined to the bottom layer and does not extend above the low-level inversion. Consequently, the only properties of the incident wave to enter the analysis of the last section were its frequency o and wavenumber K ~ , and the value of K, has only a minor influence on the properties of the solution near the escarpment.

To illustrate this and show how the analysis applies even when the incident wave is a growing baroclinic disturbance, a new example will be worked out in which another layer (denoted by suffix zero) is added at the top. For simplicity, it will be assumed to have thickness H , like the layer immediately underneath, and both these layers will be assumed to be much thicker than the bottom layer. The incident wave is now a baroclinic disturbance on a shear flow, as the velocity Or,, in the uppermost layer will, in general, be different from the velocity U , in the layer immediately below. The dispersion relation is, therefore, the one discussed by Phillips (1954) (see Pedlosky 1980, 7.1 I ) and has the same form as Eq. (19) because that is the dispersion relation for a two-layer system. The relation can be written


where ? = c-u 0 = (UO - U,>//3L2

E2 = g " H l / f 2 L 2

(35)

and g" is the reduced gravity between the upper two layers. The factor 6 in Eq. (19) is replaced by unity because the upper two layers have equal thickness.

Equation (34) is a quadratic in t whose solution is

K2(2+&2K2)(?-+0) = -(I + Z 2 K 2 ) f {02(f&2K2-E-2)2+ 1 - 02}+. (36)

For example, suppose that E ̂ = 1 so that the Rossby radius of the synoptic mode is equal to its meridional scale. Also suppose that

K~ = (1 + k,2)2 = 2 i.e. k , = 0.64 (37)

so that the incident wavelength is the one at which instability first occurs as 0 is increased upwards through its critical value, which is unity in this case. Then Eq. (36) becomes

? = 0.5(0- l )+i0.2(02-l)* . (38)

for the unstable root. The case U = 1.0, 0 = 3.6 corresponding to U , = 5ms- ' and Uo- U , = 18 ms- ' gives w = 1.5 (period 5 days) as assumed previously, and an e-folding time of 2.3 d, i.e.

w = 1.5+0.44i (39)

For the trapped wave, suppose that c2 = 0.25 in this case, corresponding to a Rossby radius for this mode of 250 km. The value of k , given by Eq. (22) is then

k , = 1.1-2.3i . (40)

The amplitude A given by Eq. (30) turns out to be

A = -0*64exp(-0*12i)

for the northgoing wave ( I = 1) and

A = - 1.54exp(0.49 i )

for the southgoing wave ( I = - 1). The value of g'h/p2 is again unity for the trapped wave, and is equal to

- 0.04 - 0.09 i

for the incident synoptic wave. Contour plots of p z and h for the new case look remarkably similar to the ones shown

in Fig. 2, the main differences being the greater zonal elongation of the incoming wave, the smaller trapping scale and the increase in amplitude of the whole pattern with time. These differences are mainly due to imposed changes rather than to any sensitivity of the trapped wave to the nature of the synoptic wave which is forcing it.

6. DISCUSSION

The main point of the present model is to have a simple representation of the incident synoptic-scale systems so their interactions with the escarpment can be illustrated. The


incident systems move purely eastwards and the escarpment has north-south orientation, so only dynamic factors can give a preference for southward propagating, coastally trapped disturbances. Such disturbances are indeed found. They are quite prominent in the interface elevation, but less obvious in the surface pressure pattern.

The model cannot produce lows which are more prominent than highs because it is linear. The effect of non-linearity in giving more prominence to lows has, however, been modelled by Gill (1977). The main respect, apart from inclusion of non-linearity and dissipation, in which the present model differs from that of Gill (1977) is in the explicit representation of the incident synoptic-scale waves, and this required the inclusion of beta- effects and a mean westerly wind above the inversion. The beta-effect does not have much influence on the properties of the trapped wave because the term paZ in Eq. (24) is less than 2ms- ' . However, the north-south slope of the interface required to balance the shear between the two layers does have a significant effect. The slope is rather larger than ob- served - an inevitable difficulty associated with the attempt to represent the situation in terms of two homogeneous layers - so perhaps the trapped wave in the present model differs from a Kelvin wave more than it should! Clearly, one would like to have a better representation of the mean flow, but this is difficult because of the presence of the escarpment which blocks the flow at lower levels. For there to be no thermal wind across the escarpment, the blocked flow has to be isentropic so, if there is a horizontal temperature gradient above the escarpment, there has to be an inversion of variable strength at the level of the plateau. This is just an indication that the properties of the inversion are likely to be bound up with the mechanics of the blocking process, but there does not seem to be any model of this as yet.

Note: The first author was awarded a J. T. Knight prize for an essay on the above topic, submitted in January 1979. This contains more details of the mathematics and a copy has been deposited in the Cambridge University Library.

Brunt, D. and Douglas, C. K. M.

Gill, A. E.

Gill, A. E., Green, J. S. A. and Simmons, A. J.

LeBlond, P. H. and Mysak, L. A.

Longuet-Higgins, M. S.

Pedlosky, J. Phillips, N. A.

Preston-Whyte, R. A. and

Preston-Whyte, R. A., Diab, R. D. Tyson, P. D.

and Tyson, P. D.

1928

1971

I974

977

968

980 1954

1973

1977

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pressure distribution, and its effect on rainfall, Mem. R . Met. Soc.. 3. 29-51.

Coastally trapped waves in the atmosphere, Quart. J. R . Met. Soc., 103, 431-440.

Energy partition in the large scale ocean circulation and the production of mid-ocean eddies, Deep-sea Res., 21,

Trapped coastal waves and their roles in shelf dynamics. In The Sea, Vol. 6, E. D. Goldberg, I. M. McCave, J. J. O'Brien and J. H. Stele, eds., Wiley-Interscience, New York, pp. 459495.

On the trapping of waves along a discontinuity of depth in a rotating ocean, J. Fluid Mech., 31, 417434.

Geophysical Fluid Dynamics, Springer, New York. Energy transformations and meridional circulations asso-

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Documents

Generation of coastal lows by synoptic-scale waves