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Coastally trapped disturbances in the lower atmosphere Reason, Christopher James Charles 1989

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C O A S T A L L Y TRAPPED DISTURBANCES IN T H E LOWER ATMOSPHERE by CHRISTOPHER JAMES CHARLES REASON M.Phil., The City University, London, 1985 M.Sc, University of British Columbia, 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES APPLIED MATHEMATICS/GEOGRAPHY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1989 © Christopher James Charles Reason, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of GBOG&APHI The University of British Columbia Vancouver, Canada Date / / $"/ ft  DE-6 (2/88) Abstract Coastally trapped disturbances that propagate in the marine layers of western North America, Southern Africa and southeastern Australia are examined. These areas of the world are considered to be most favourable for the propagation of the disturbances because they all possess pronounced subsidence inversions and barrier-like coastal mountain ranges. Trapping of the disturbance energy within a coastal zone then occurs through this inversion being situated below the mountain crests and through Coriolis effects on the propagating disturbances themselves. Coriolis effects are also responsible for the propagation occurring with the coast on the right (left) in the Northern (Southern) Hemisphere. • As this propagation occurs, marked changes in the inversion height and local weather conditions below the inversion are observed. These changes are similar in all three regions with the exception that the inversion is raised in the North American and Australian cases but lowered for the South African disturbances. This difference is shown to arise because the forcing flow is on- or alongshore for the former but offshore in Southern Africa. It is argued that the fundamental dynamics of these disturbances are identical (hydrostatic and semigeostrophic) in each area but that regional differences in the forcing and boundary conditions are responsible for the various manifestations of the disturbances. Based on the observed commonality between the three theory of coastally trapped disturbances is developed from the shallow water equations for a rotating, stratified and flat-bottomed fluid. It is shown that the theory will admit two types of solution, a Kelvin wave and a coastal gravity current, which if higher order effects are included, are found to be related. Comparisons of the different forcings and boundary conditions are made to show the potential importance of nonlinearities. It is concluded that the Southern African case is best described as a continuously forced, linear Kelvin wave, while the North American and Australian disturbances exhibit both gravity current and nonlinear Kelvin wave characteristics. In each case, the theoretical predictions of the evolution time scale, propagation characteristics and speed are shown to be consistent with the available observations. IV Table of Contents Abstract ii Table of Contents iv List of Tables viii List of Figures ix Acknowledgements xxii Chapter 1. Introduction 1 Chapter 2. A general theory of coastally trapped disturbances. 10 2.1. Dynamic features common to all coastally trapped 10 disturbances. 2.2. Governing equations for coastally trapped disturbances. 17 2.3. Differences between coastally trapped gravity currents 28 and Kelvin waves: 2.4. Some forced solutions to the shallow water equations. 35 2.5. Nonlinear semigeostrophic theory of coastally trapped 44 disturbances. 2.6. Scale parameters characterising the dynamics of 53 coastally trapped disturbances. 2.7. Variations in local boundary conditions, synoptic 63 forcing and propagation characteristics of coastally trapped disturbances. Chapter 3. Coastal ridges of California and the Pacific 71 North West. 3.1. Introduction 71 3.2. Synoptic description of the Californian event 71 •'• of 3-7 May,.1982. 3.3. Solitary Kelvin wave or gravity current ? 83 3.4. Solitary Kelvin wave model of the May, 1982 event. 98 vi 3.5. Conclusion 107 Chapter 4. The Southern African coastal low. 110 4.1. Introduction 110 4.2. Synoptic conditions. 115 4.3. Atmospheric Sounding Data. 120 4.4. Comparison of coastal low behaviour with Kelvin 129 waves. 4.5. The contribution of berg winds to coastal low 133 forcing. 4.6. Conclusion 138 Chapter 5. The coastal ridges of Southeast Australia. 142 5.1. Introduction 142 5.2. Synoptic conditions during the November, 1982 145 event. vii 5.3. Application of the modified Kubokawa and Hanawa 152 model. 5.4. Conclusion 163 Chapter 6. Summary and Conclusions. 166 References 175 Appendix 1. Wave evolution equation for dissipative case. 18.1 Appendix 2. Other North American coastally trapped.events. 185 A2.1. Californian event of 13-20 July, 1982. 185 A2.2. California to Vancouver Island event of 201 May 14-17th, 1985. List of Tables Table 2.1 - page 14: Rossby radii, alongshore length scales, and time scales for the coastally trapped disturbances of Southern Africa, Australia and North America. Table 2.2 - page 55: Rossby (Ro) and Froude (Fr) numbers and ratio (Ro/Fr) for the coastally trapped disturbances of Southern Africa, Australia and North America. Table 2.3 - page 62: Semigeostrophic parameters for the coastally trapped disturbances of Southern Africa, Australia, California and the Pacific Northwest (PNW). Table 2.4 - page 64: Topographic dimensions and Rossby and Froude numbers of the incident flow for the coastally trapped disturbances of Southern Africa, Australia and North America. Table 2.5 - page 70: Continentality indices Table 4.1 - page 130: Observed propagation speed and computed speeds for the Southern African coastal low. Table 5.1 - page 155: Evolution times and speeds of propagation of the Australian coastal ridge. List of Figures Figure 1.1 - page 3: World map showing the subtropical location of coastally trapped disturbances. Figure 1.2 - page 4: Conceptual view of the propagation of a coastally trapped disturbance in the marine layer showing the relation of this layer to the coastal mountains. Figure 2.1 - page 20: Topographic map of Southern Africa. The first contour inland from the coastline is the 500 m contour, that enclosing the stippled region is the 1000 m contour and that enclosing the shaded region is the 1500 m. AB = Alexander Bay, CT = Cape Town, PE = Port Elizabeth,D = Durban. On the vertical axis, the latitudes are in degrees south whereas on the horizontal, the longitudes are in degrees east. Figure 2.2 - page 21: Topographic map of Australia. The unshaded region refers to elevations of 0 - 300m, the shaded region to 300 - 900m and the black area to topography over 900m. Adapted from Holland and Leslie (1986). Figure 2.3 - page 22: Topographic map of western North America. The unshaded region refers to 0 - 300m, the light stippled region to 300 - 1500m and the dark shaded area to land at elevations greater than 1500 m. Adapted from Mass and Albright (1987). Figure 2.4 - page 23: Schematic showing the model topography and atmospheric stratification. X Figure 2.5a - page 31: Perspective view of a coastally trapped gravity current in the Californian marine layer showing the position of the curved density front x = -L'(y,t). Note that the displacement of the gravity current is maximum near the coastal mountains and is zero at the density front, i.e., at the front the marine layer depth equals its mean value of 400 m. Adapted from Dorman (1987). Figure 2.5b - page 31: Side view of a typical atmospheric gravity current showing the greater depth of the head than the feeder flow behind. Adapted from Seitter and Muench (1985). Figure 3.1 - page 72: GOES-WEST visible satellite imagery of the southwest U.S. and adjacent Pacific Ocean for 23 UTC May 3 to 15 UTC May 7, 1982. Adapted from Mass and Albright (1987). Figure 3.2a - page 73: Topographic and place location map of the west coast of the U.S. and southern British Columbia, Canada. Adapted from Mass and Albright (1987). Figure 3.2b - page 74: Topographic and meteorological station location map of California and southern Oregon. Note that only the last two digits of each buoy number (as in the text) are given unlike in Fig. 3.2a in which all five digits are given. Adapted from Dorman (1985). Figure 3.3 - page 76: National Meteorological Center 850 mb synoptic maps for the period 00 UTC May 3 to 00 UTC May 8, 1982. Geopotential heights (solid) are XI in decametres and isotherms (dashed) are in degrees Celsius. Adapted from Mass and Albright (1987). Figure 3.4a - page 77: Mesoscale sea level pressure analyses and surface winds for the period 00 GMT May 4 to 18 UTC May 7, 1982. Solid lines are sea level isobars (lOxx mb) and winds are in knots. Heavy dashed lines are the northern boundaries of the coastal stratus as defined by satellite visible imagery. Adapted from Mass and Albright (1987). Figure 3.4b - page 78: National Meteorological Center surface synoptic maps for the period 00 UTC May 3 to 00 UTC May 8, 1982. Solid lines are sea level isobars (lOxx mb). Adapted from Mass and Albright (1987). Figure 3.5a - page 79: Temperature (right hand curve) and dewpoint temperature (left hand curve) soundings through the lower atmosphere for San Diego for the period 00 UTC May 2 to 12 UTC May 7, 1982. The base of the inversion is indicated by an arrow on each sounding. Figure 3.5b - page 80: Temperature (right hand curve) and dewpoint temperature (left hand curve) soundings through the lower atmosphere for Point Mugu for the period 11 UTC May 3 to 11 UTC May 7, 1982. The base of the inversion is indicated by an arrow on each sounding. . Figure 3.5c - page 82: Temperature (right hand curve) and dewpoint temperature (left hand curve) soundings through the lower atmosphere for Vandenburg for the period 00 UTC May 2 to 12 UTC May 7, 1982. The base of the inversion is indicated by an arrow on each sounding. Figure 3.6 - page 84: Height of the base of the inversion at Vandenburg and Oakland for the period 00 UTC May 2 to 12 UTC May 8, 1982. The inversion lifted and descended at Oakland 24 hours after it did at Vandenburg. Adapted from Dorman (1985). Figure 3.7 - page 84: Surface winds and pressures for Gualala station and offshore buoys (C3, C5) on the central Californian coast during the May, 1982 event. The winds have been rotated so that "North" is actually 317° which is aligned with the coast. Station Sea Ranch is located on the coast a few kilometres south of Gualala. Adapted from Dorman (1985). Figure 3.8a - page 85: Surface pressure traces for various stations along the central Californian coast for the May, 1982 (left hand plot in each case) and July, 1982 (right hand plot) events. On the vertical axis of each trace, the numbers (xx) refer to lOxx millibars while the horizontal axis gives the day in either May or July. Figure 3.8b - page 86: Alongshore surface pressure differences (in millibars) for various stations along the central Californian coast for the May, 1982 (left hand plot in each case) and July, 1982 (right hand plot) events. In each plot, the horizontal axis gives the day in May or July. 1 = buoys 22 (not shown, but located 50 km north of buoy 14, see Fig 3.2a) - 14, 2 = buoys 14 -13,3 = Point Arena - Fort Ross, 4 = buoys C5 - 13. Note that 1 (obtained from Dorman (1985)), is plotted with the scale on the vertical axis reversed and that this plot as well as plots 2-4 are such that a positive pressure difference corresponds to a northerly wind for downgradient flow. Figure 3.9 - page 89: Surface winds at stations (see Fig. 3.2b for locations) along the Californian coast. Wind speeds are given in m/s and dots indicate calm. See legend for wind barbs in m/s: a) 1, b) 5, c) 2.5, d) 12.5, e) gusting to 10, f) 7.5 m/s gusting to 15 m/s, g) calm, h) a wind from the north and from the southwest. Adapted from Dorman (1985). Figure 3.10 - page 90: Surface pressure and winds along the northern Californian coast (just south of Cape Mendocino) recorded at 1800 UTC, July 17, 1982. The leading edge of the gravity current has reached Point Arena where significant across-shore winds are observed. The wind vectors point downstream. Adapted from Dorman (1987). ' Figure.3.11 - page 91: Surface temperature, winds and pressure recorded at Astoria, Oregon during the May, 1985 gravity current event. Significant across-shore (NE and W) winds are observed. Adapted from Mass and Albright (1987). Figure 3.12 - page 92: Mesoscale sea level pressure analyses and surface winds for the May, 1985 event. The heavy dashed line marks the leading edge of the gravity current. With the exception of the first and last panels, significant across-shore winds are observed near each leading edge. Adapted from Mass and Albright (1987). Figure 3.13 - page 93: GOES-West satellite visible imagery for 0045 UTC May 3, 1982. The localised initial displacement (low level stratus deck) of the marine layer in the Southern Californian Bight is evident. Note the trapping by the coastal mountains. Courtesy of Dr. Dorman. XIV Figure 3.14 - page 93: GOES-West satellite visible imagery for 2315 UTC May 3, 1982. Some erosion of the stratus cloud due to diurnal heating is evident on comparison of the stratus deck with that shown in Fig. 3.13. Note that UTC is 7 hours ahead of local time. Courtesy of Dr. Dorman. Figure 3.15a - page 95: GOES-West satellite visible imagery for 1616 UTC July 12, 1982. Note the reservoir formation (widespread stratus cloud) in the marine layer of the Southern Californian Bight and the trapping of the northern propagating part of the stratus to a narrow coastal zone. The leading edge of this stratus deck has reached Monterey Bay just south of San Francisco. Courtesy of Dr. Dorman. Figure 3.15b - page 96: GOES'-West satellite visible imagery for 2115 UTC July 12, 1982. The much wider spread displacement (indicated by stratus cloud deck) of the marine layer (and hence reservoir formation) in the Southern Californian Bight than for Figs. 3.13-3.14 is evident. Courtesy of Dr. Dorman. Figure 3.16 - page 101: GOES-West satellite visible imagery for 1515 UTC May 4, 1982 showing the propagation of the stratus deck to Monterey Bay south of San Francisco (see Fig. 3.2a for location of place names) about 100 km north of the Southern Californian Bight. Courtesy of Dr. Dorman. Figure 3.17 - page 101: GOES-West satellite visible imagery for 1615 UTC May 5, 1982 showing the separation of the stratus overcast from the coast at Cape Mendocino (upper arrow). Eddy formation here and at Point Arena (lower arrow) is evident. Adapted from Dorman (1985). X V Figure 4.1 - page 112: Topographic and location map of Southern Africa. AB refers to Alexander Bay, CT to Cape Town, CA to Cape Agulhas, PE to Port Elizabeth, and D to Durban. The first contour inland from the coastline is the 500 m contour, that enclosing the stippled region is the 1000 m contour, and that enclosing the black area is the 1500 m contour. Figure 4.2 - page 116: Synoptic pressure maps for the surface at 14 00 UTC daily obtained from the South African Weather Bureau. The map borders for each panel are 20-45 S, 5-45 E. In each figure, H refers to the ridging anticyclone, the bold line to the trailing mid-latitude frontal system and the arrow to the coastal low. Case 1 extends over 6-11 February, 1981; Case 2 over 15-20 September, 1985 and Case 3 over 18-23 April, 1980. Pressures over the sea are given as the difference in millibars from 1000 mb and the isobars are contoured at 2 mb intervals. Over the continent, the isolines refer to the 850 mb surface contoured at 10 gpm intervals. Figure 4.3 - page 117: Time-height plots (4.3a refers to Case 1, 4.3b to Case 2 and 4.3c to Case 3) of wind speed (contoured at 4 m/s interval) and direction (arrows, where the point of the arrow refers to the time-height co-ordinates of the datum). Wind speeds greater than 8 m/s are stippled. The dashed line refers to the minimum wind speeds recorded during each event. On the time axis, 0 refers to the time of coastal low arrival, 12 to 12 hours after arrival and -12 to 12 hours before arrival, etc. On the vertical axis, the pressures are in millibars and the labels AB, CT, etc., are as in Fig. 4.1. Figure 4.4 - page 121: Temperature (right hand curve) and dewpoint temperature (left hand curve) profiles through the lower atmosphere. Fig. 4.4a refers to Case 1, XVI 4.4b to Case 2 and 4.4c to Case 3. On the vertical axis, the pressure level p in the atmosphere is given as 1020 - p millibars. The height of the inversion base is indicated by the arrows. On the vertical axis, the labels AB, CT, etc. are as in Fig. 4.1. Figure 4.5 - page 128: Time-height variability of the geopotential anomalies (gpm) from the mean recorded during the event. Fig. 4.5a refers to Case 1, 4.5b to Case 2 and 4.5c to Case 3. Anomalies below -30 gpm are shaded. Labels on the horizontal axis are as in Fig. 4.3. On the vertical axis, pressures are in millibars and the labels AB, CT etc. are as in Fig. 4.1. Figure 4.6 - page 137: The free (unforced) linear Kelvin wave (upper curve) and berg wind forced linear Kelvin wave (lower curve) model solutions. Figure 5.1 - page 143: Topographic and station location map of Australia! Unshaded regions are at 0-300 m elevation, stippled regions at 300-900 m elevation and dark areas at greater than 900 m elevation. Adapted from Holland and Leslie (1986). Figure 5.2 - page 144: Japanese geostationary satellite visible imagery for - a) 00 UTC November 10, 1982 showing the cold outbreak (signified by the narrow cloud band) at the eastern end of the Great Australian Bight and - b) 00 UTC November 11, 1982 showing the stratiform cloud associated with the propagating coastal ridge on the east coast of Australia. Adapted from Holland and Leslie (1986). Figure 5.3 - page 146: Mean sea level isobaric analyses for 00 UTC, November 10, 1982 (Fig. 5.3a) and 00 UTC, November 11, 1982 (Fig. 5.3b). The forcing anticyclone. in the Great Australian Bight is labelled H. To the east of this anticyclone, over the Tasman Sea, is a short wave trough and frontal zone. Isobars are in 4 millibar intervals. Adapted from Holland and Leslie (1986). Figure 5.4 - page 148: Vertical cross-sections of potential temperature (K) at various stations along the east Australian coast for November 9, 1982 (Fig. 5.4a), November 10, 1982 (Fig. 5.4b) and November 11, 1982 (Fig. 5.4c). The stations used and the observation times are indicated on the abscissa. Adapted from Holland and Leslie (1986). Figure 5.5 - page 150: Time-height variability of winds and potential temperatures (K) at Williamtown for 9-11 November, 1982. On the wind arrows, one full feather indicates 5 m/s. Adapted from Holland and Leslie (1986). Figure 5.6 - page 151: Barograph traces of surface pressure for various inland and coastal stations in eastern Australia. The pressure traces cover the period 00 UTC, November 9 to 12 UTC, November 11, 1982 and the tick marks are at 10 mb spacing. The unshaded region refers to terrain at 0-600m elevation, the stippled to terrain of 600-1000m and the black region to terrain higher than 1000m. Adapted from Holland and Leslie (1986). Figure 5.7 - page 162: Steady state, undular bore solution of the Korteweg- . deVries-Burgers equation. The normalised variables z and $ are proportional to . interface displacement and wave front propagation distance respectively. Adapted from Whitham (1974). Figure A l - page 186: The geopotential height (metres) of the 700 mb surface in the atmosphere on July 16, 1982 at 1200 UTC. Note the trough, which forced the July, 1982 event, crossing the coast. Adapted from Dorman (1987). Figure A2a - page 187: Temperature (right hand curve) and dewpoint temperature (left hand curve) soundings for San Diego during the period 00 UTC July 12 to 00 UTC July 23, 1982. The base of the inversion is indicated by an arrow on each sounding. Figure A2b - page 189: Temperature (right hand curve) and dewpoint temperature (left hand curve) soundings for Point Mugu during the period 11 UTC July 12 to 23 UTC July 16, 1982. The base of the inversion is indicated by an arrow on each sounding. Figure A 3 - page 191: Surface pressure and winds for July 17, 1982 at 1200 UTC. The hatched area denotes the zone of raised marine layer corresponding to the propagating gravity current. Note the across-shore winds just behind the leading edge of the gravity current. Numbers are the surface pressure in millibars minus 1000 and multiplied by 10. Isobars are labelled in millibars. Adapted from Dorman (1987). Figure A4 - page 192: Surface pressure and winds along the northern Californian coast (just south of Cape Mendocino) on July 17, 1982 at 1800 UTC. The leading edge of the gravity current has reached Point Arena where significant across-shore winds are observed. Wind vectors point downwind. Adapted from Dorman (1987). Figure A 5 - page 194: Base height of the inversion recorded at coastal sounding stations. Adapted from Dorman (1987). xix Figure A 6 - page 195: Surface pressure and winds for July 16, 1982 at 1800 UTC. The hatched area denotes the zone of raised marine layer corresponding to the propagating gravity current. Numbers are the surface pressure in millibars minus 1000 and multiplied by 10. Isobars are labelled in millibars. Adapted from Dorman (19.87). Figure A 7 - page 199: GOES-West satellite visible image for 1615 UTC July 19, 1982. The arrow indicates Cape Blanco, which is the northernmost extent of the raised marine layer and stratus overcast. Adapted from Dorman (1987). Figure A 8 - page 202: Mesoscale sea level pressure analyses and surface winds for the period 12 UTC May 16 to 12 UTC May 17, 1985. The heavy dashed line indicates the leading edge of the gravity current. Adapted from Mass and Albright (1987). Figure A 9 - page 203: Soundings of temperature (solid lines) and dewpoint temperature (dashed lines) for San Diego over the period 12 UTC May 14 to 12 UTC May 16, 1985. Winds (knots) are plotted to the right of each sounding. The base of the inversion is indicated by arrows on each sounding. Adapted from Mass and Albright (1987). Figure A10 - page 204: GOES-West satellite visible imagery for the period 21 UTC May 15 to 23 UTC May 17, 1985. The stratus deck associated with the propagating coastal ridge is evident. Adapted from Mass and Albright (1987). X X Figure A l l - page 205: Mesoscale sea level pressure analyses for the period 12 UTC May 15 to 18 UTC May 17, 1985. Heavy dashed lines are the northern boundaries of the coastal stratus, as determined from the satellite imagery. Isobars are lOxx mb and observations over the ocean are indicated by solid dots. Adapted from Mass and Albright (1987). Figure A12 - page 207: Surface temperature, pressure and winds recorded at Astoria, Oregon during the period 20 UTC May 16 to 12 UTC May 17, 1985. Significant across-shore winds (NE and W) occur particularly near the time (0100 UTC May 17) that the leading edge of the coastal ridge reached this station. Adapted from Mass and Albright (1987). Figure A13 - page 208: Temperature (solid lines) and dewpoint temperature (dashed lines) soundings for Oakland over the period 12 UTC May 14 to 12 UTC May 16, 1985. Winds (knots) are plotted to the right of each sounding. The base of the inversion is indicated by the arrow on each sounding. Adapted from Mass and Albright (1987)! Figure A14 - page 209: Propagation speed of the leading edge of the coastal ridge based on coastal wind changes (solid lines) and the northern boundary of the coastal stratus (crosses). Also shown are the theoretical gravity current speeds calculated from the formula of Seitter and Muench (1985). Adapted from Mass and Albright (1987). Figure A15 - page 209: Temperature (solid lines) and dewpoint temperature (dashed lines) soundings at Quillayute, Washington before (00 UTC May 17, 1985) and after (12 UTC May 17,-00 UTC May 18, 1985) the passage of the leading edge XXI of the ridge at that location. Winds (knots) are plotted to the right of each sounding. The base of the inversion is indicated by the arrow on each sounding. Adapted from Mass and Albright (1987). Figure A16 - page 211: Surface winds recorded at various coastal stations (see Figs 3.2a-b for locations of place names). Solid dots indicate three hourly observations and dotted lines indicate missing data. Tick marks represent 5 knot intervals. Adapted from Mass and Albright (1987). Figure A17 - page 212: Microbarograph traces of surface pressure recorded at various coastal stations from 08 UTC May 16 to 18 UTC May 17, 1985. The vertical arrows indicate times of passage of the leading edge of the coastal ridge. The dashed horizontal lines indicate the 1012 mb pressure level and the tick marks on the vertical axis are at 1 mb intervals. Adapted from Mass and Albright (1987). Acknowledgement Firstly, I should like to thank my thesis advisor Dr. D.G. Steyn and the other members of my supervisory committee, Drs. P.H. LeBlond, T.R. Oke and B.R.Seymour, for the invaluable advice and guidance that they have extended to me during the course of my study. I am very grateful to Dr. M.R. Jury and Professor G.B. Brundrit of the Department of Oceanography, University of Cape Town for providing a most supportive environment for me to work in whilst I was collecting and analysing data for the Southern African coastal low case study. In a similar vein, I should like to express much gratitude to Dr. C E . Dorman of San Diego State University who was most generous in giving both advice and unanalysed data. I would also like to thank Dr. C F . Mass of the University of Washington for the use of some of his satellite images. Much appreciation is due to Sheryl Tewnion who produced many of the computer plots. Finally, I should like to acknowledge the financial support received from the NSERC research grant of my advisor, Dr. D.G. Steyn, for funding for research over two summers at UBC and which enabled me to to visit Dr. C E . Dorman in San Diego. 1 Chapter 1. Introduction Thermal and orographic influences on atmospheric motion are often dominant in driving the flow away from a geostrophic balance. Many meteorological phenomena arise through the action of one or the other of these forces and it is to be expected that in situations where both are important, the resulting atmospheric dynamics will be particularly interesting. A good example of such a situation is offered by the meteorology of those coastal areas that are bordered by substantial mountain ranges (where by substantial it is meant an increase in elevation of 1 km or more extending along the coast for hundreds of kilometres). Such a topographic barrier is large enough to influence atmospheric motion on both the meso- and the synoptic scales and its effect in perturbing low level winds induced by either local contrasts in temperature between land and ocean or by the large scale pressure gradient is anticipated to lead to the forcing of vigorous flows in the coastal region. Several dramatic examples of propagating changes in mesoscale wind fields and local weather conditions have been observed in the vicinity of the coastal mountains of southeastern Australia, South Africa, Chile and the Pacific seaboard of North America (Gill, 1977; Anh and Gill, 1981; Bannon, 1981; South African Coastal Low Workshop - hereafter referred to as CLW, 1984; Dorman, 1985, 1987; Holland and Leslie, 1986; Mass and Albright, 1987). The mechanism by which these local weather changes are brought about involves the propagation along the coast of low level disturbances. These disturbances are trapped against the coastal mountains and result in the replacement of the warmer, drier air ahead of the disturbance by cooler, moister air. Overhead, but at a level usually just below the mountain crests, the propagation of these phenomena is invariably accompanied by banks of stratus clouds. Surface pressure gradients are always 2 perturbed to some extent by these disturbances; in the case of southeastern Australia and western North America coastal ridges of high pressure are usually formed whereas in South Africa coastal lows are common. The topographic trapping is an important feature of the dynamics because it limits the width of the disturbance and cloud deck to typically within 100-300 km of the coastal mountains. Either dissipation or interaction with the large scale mean wind ensures that the propagation of these disturbances along the coast does not, as a rule, continue much beyond 500-1000 km. Both this alongshore scale and the offshore one of 100-300 km characterise the disturbances as being mesoscale. Figure 1.1 indicates the tendency for these mesoscale phenomena to be found along those subtropical coastal areas that, in addition to being bounded by a substantial mountain barrier, are also subject to semi-permanent subsidence. The latter arises from the almost continual presence of a large anticyclone over the adjacent oceans (e.g., the North Pacific and South Atlantic Anticyclones) and results in a marine inversion layer being found along these coasts. By preventing the leakage of energy upwards, the. inversion layer acts as a horizontal waveguide to allow the propagation of mesoscale disturbances along the coast. Lateral trapping of the disturbances within a coastal zone, always much less in width than the distance alongshore that propagation occurs, is then affected by the action of the Coriolis force on the flow along the mountain barrier. It must be emphasized that the mountain barrier must be at least as high as the inversion layer in the coastal atmosphere for effective trapping to occur and that the flow normal to the coastal mountains must vanish at the mountains themselves. Figure 1.2 provides a conceptual view of the typical atmospheric and topographic conditions associated with the propagation of the disturbances examined in this thesis. 3 Figure 1.1. World map showing the subtropical location of coastally trapped disturbances. 4 Upper a t m o s p h e r e S u b s i d e n c e i n v e r s i o n p r o p a g a t i o n speed a l o n g s h o r e l e n g t h s c a l e n o r t h e r l y w i n d ( s u r f a c e ) s o u t h e r l y w i n d ( s u r f a c e ) V C o a s t a l Mounta i n s M a r i n e L a y e r N S \ ^ ^ Ocean Land Figure 1.2. Conceptual view of the propagation of a coastally trapped disturbance in the marine layer showing the relation of this layer to the coastal mountains. 5 Strengthening of the subsidence inversion and thus more effective trapping of the disturbance may occur via local sea breezes, that are forced by land/sea temperature differences and which lead to an increase in the stability of the atmospheric layer below the inversion (hereafter referred to as the marine layer), Holland and Leslie (1986). On a larger scale, e.g. the synoptic, land/sea temperature differences may provide thermal forcing for the disturbances,in addition to the topographic effects already discussed (Chao, 1985; Coulman et al., 1985). In principle, it is not necessary for the disturbances to be confined to coastal areas, because all that is required is a substantial mountain range to present a reasonably smooth barrier to the incident synoptic flow and an inversion to trap the resulting energy below the mountain tops. Thus, low-level circulation features have been observed^prppagating around major mountain ranges such as the Rockies in North America and those in northwestern China and eastern Siberia which are roughly normal to the synoptic westerlies (Hsu, 1987). In general, these features tend to be on a larger scale than the coastally trapped disturbances and, although the fundamental dynamics of the two phenomena are essentially similar, the former will not be dealt with further in this thesis. Two types of atmospheric dynamics have been proposed to explain coastally trapped disturbances. Firstly some form of Kelvin wave has been suggested for the coastal lows of South Africa (Gill, 1977; Anh and Gill, 1981; Bannon, 1981), the coastal ridging of southeastern Australia (Holland and Leslie, 1986) and for the May 1982 Californian case (Dorman, 1985). Secondly, topographically trapped gravity currents have been hypothesised for the surges of California and the Pacific North West (Dorman, 1987; Mass and Albright, 1987) and for the Southerly Buster of Australia (Baines, 1980). Note that Pacific North West refers to Oregon, Washington and British Columbia and is hereafter abbreviated as PNW. 6 Although in his theoretical model Baines (1980) considered the Southerly Buster to be topographically trapped, later observational studies (Colquhoun et al., 1985; Coulman et al., 1985) have indicated that this is often not the case. Instead, the Southerly Buster appears to propagate northwards along the coast in a squall-like fashion as an almost linear front, exhibiting little or none of the offshore decay of alongshore velocity and marine layer displacement characteristic of a topographically trapped phenomenon. Thus, the Baines gravity current model, and for exactly the same reasons a Kelvin wave model, would not be appropriate for this type of propagation. In a more recent study, Holland and Leslie (1986) have reported that the Southerly Buster often occurs at the leading edge of a ridging surge of high pressure that propagates northwards along the southeast Australian coast. This coastal ridge is thought to form initially as a trapped gravity current which then evolves via a superimposed Kelvin wave that propagates through and ahead of the gravity current. In agreement with Holland and Leslie (1986), the coastal ridging with its observed offshore decay of alongshore velocity and marine layer displacement is considered here to be a coastally trapped disturbance. On the other hand, Southerly Busters, which are of smaller scale and shorter timespan (less than an inertial period usually) than the coastal ridges, do not exhibit any significant decay in alongshore velocity or displacement away from the coast, and which are squall-like in nature, are not considered coastally trapped although they are certainly constrained by the coastal mountains (Colquhoun et al., 1985). Further support for the lack of trapping by the coastal mountains is evident from the numerical simulations and observations of Howells and Kuo (1988). Indeed, when the topography was set to zero in their model, these authors found little change in the simulated structure and propagation 7 "of the Buster. Rather, it was found that the local land-sea temperature and frictional contrasts controlled the evolution of this phenomenon. On the basis of these results, it may be concluded that the Southerly Buster, which was in earlier studies (Baines, 1980; Colquhoun et al., 1985) thought to be similar in dynamics to the Southern African coastal low, is in fact not a coastally trapped disturbance in the sense defined here and as such is not discussed further in this thesis. As hinted above for the Australian coastal ridges, and as will be discussed later, the Kelvin wave and gravity current models share several common features. With this in mind, it is important that any study of mesoscale coastally trapped disturbances be fundamentally concerned with understanding what type of dynamics is most appropriate. Particular attention in this thesis will therefore be devoted to this end. The approach that will be taken here will involve the consideration of the essential similarities of the disturbances in each area to provide a general theory of coastally trapped disturbances and then an investigation into the different forcings and boundary conditions of each region in an attempt to understand why the disturbances take the various forms they do. Underpinning this analysis will be the theme that, despite the obvious and sometimes considerable differences between the coastally trapped disturbances of each area, the fundamental dynamics are essentially the same. These dynamics are, however, often considerably modified and perhaps, even controlled by the regional conditions. An example of the latter would be the effect of Cape Mendocino in preventing further propagation northwards of the. Californian event of May, 1982 (Dorman, 1985; Reason and Steyn, 1988). Existing studies of coastally trapped disturbances tend to lack clarity on several fundamental points. Most importantly, the lack of distinction in the possible interpretation of the underlying dynamics of the Australian and North American cases as being of a gravity current or Kelvin wave nature needs to be resolved as far 8 as the available data will allow. This task will be attempted in the subsequent chapters. Note that in the case of the South African coastal low however, there seems little doubt that this disturbance is a Kelvin wave (Gill, 1977; Anh and Gill, 1981; Bannon, 1981; CLW, 1984). Another aspect which will be examined further is the generation mechanism of the disturbances. In all cases, some form of synoptic scale disturbance interacting with the coastal mountains appears to force the disturbance but the details of how this mechanism works are not well known. Significant differences between the various cases also exist. For example, the synoptic forcing in South Africa induces a warm, offshore flow ahead of the coastal low which will be shown to be crucial for the formation of a low rather than a high. On the other hand, the pre-disturbance flow of the coastal ridges of Australia and North America is generally on- or alongshore. The effect of variable surface and boundary conditions along the propagation path of the disturbance will also be investigated in more depth. Examples of such effects are local land-sea temperature differences and variations in sea surface temperature, surface friction and in the local topography. The latter often plays an important role in determining the local speed and coherency of propagation of the disturbance and if it varies significantly over a large enough distance may even cause cessation (Dorman, 1985; Reason and Steyn, 1988). Previous theoretical studies (Gill, 1977; Anh and Gill, 1981; Bannon, 1981) have not modelled the effects of such variations explicitly. Apart from the intrinsic interest to coastal meteorology, further motivation for the research can be found in the practical applications to forecasting the complex weather changes of these regions. Due to the spatial scale of coastally trapped 9 disturbances being generally too small to be accurately resolved by standard weather networks, the forecasting of these phenomena has always been difficult. Also, the weather changes associated with these disturbances are usually energetic and occur quickly. In all areas, the arrival of the disturbance is usually accompanied by rapid shifts in wind direction, large increases in wind speed (to 15 m/s or more) and temperature drops of typically 10-15 °C within a few minutes as warm, subsiding continental air is replaced by cooler, moister maritime air. Additionally, there is often a transition from nearly cloud-free to fog or overcast stratus conditions. The implications of variations in local weather of this magnitude for coastal shipping and aviation are obvious. 10 Chapter 2. General Theory of Coastally Trapped Disturbances 2.1. Dynamic features common to all coastally trapped disturbances The essential feature common to all the marine layer disturbances examined in this thesis is, as the chapter heading implies, the topographic trapping, via the Coriolis force, of the disturbance to within a distinct zone seawards of the coastal mountains. Previous studies (Gill, 1977; Anh and Gill, 1981; Bannon, 1981; CLW, 1984; Dorman, 1985, 1987; Holland and Leslie, 1986; Mass and Albright, 1987) have identified this distinct trapping zone with the internal Rossby radius characteristic of the particular coastal atmosphere in which the disturbance is propagating. Justification for this identification is on the basis of the good agreement between the observed width of the disturbances studied by these authors with the computed Rossby radius and on consideration of the governing dynamics of the disturbances. For example, it is well known that the Rossby radius is a fundamental length in the theory of rotating, stratified fluids which measures the scale at which the effects of the Coriolis force become as important as the stratification. It is easily shown (see below and standard texts like Gill (1982) and Pedlosky (1987)) that the Rossby radius emerges as the e-folding cross-shore scale of the general coastally trapped solution to the shallow water equations of motion. Following on from this result is the fact that the alongshore velocity and interfacial or inversion displacement decay exponentially in magnitude from the constraining side boundary of the fluid. The observation in all the North American, Australian and South African cases, i.e. those cited above, that the marine layer disturbance is always constrained within a definite zone of the order of the Rossby radius in width offshore from the coastal mountains, is therefore powerful evidence that these disturbances are some form of Kelvin wave. 11 Another crucial property of Kelvin dynamics, which is also always satisfied by the above mentioned observations of the disturbances, is that the wave should propagate with the side boundary on the right (looking downstream) in the Northern Hemisphere and on the left in the Southern Hemisphere (LeBlond and Mysak, 1978; Gill, 1982). Note that the side boundary need not necessarily be the coastal mountains for an atmospheric wave or the coast for a oceanic wave. For example, trapped waves of Kelvin form have been observed propagating along interior mountain ranges such as the Rockies (Hsu, 1987), while in the ocean they are also found travelling along ridges and escarpments of the deep ocean floor (e.g., LeBlond and Mysak, 1978). Equatorially trapped Kelvin waves in both the ocean and the atmosphere, which are bound by the vanishing of the Coriolis force at the equator, are obviously different again and fall outside the scope of the discussion. While coastal mountains provide lateral ducting for marine layer disturbances, vertical trapping occurs because, in every case, a temperature inversion exists above the marine layer but below the mountain crests. Thus in the coastal zone, cool and dense maritime air in the lower atmosphere (i.e., the marine layer) is separated by the inversion layer from warmer, less dense continental air above. Any disturbance in either the marine or the inversion layer is prevented from propagating upwards by the weak density stratification (potential temperature roughly constant) above (Gill, 1977) so that vertical trapping of the energy occurs. Hence, the combination of the coastal mountains and the inversion acts as an efficient waveguide for the along-shore propagation of disturbances. As will be seen later, this waveguide effect is present in all observed cases. Since the disturbances generally propagate in the marine layer in a reasonably coherent fashion over a span of several days and have typical horizontal dimensions of at least a hundred kilometres, a large scale, semi-permanent inversion is required 12 to support them. The most obvious examples are the subsidence inversions of the subtropical oceans and coastal areas that result from the large scale semi-permanent anticyclones (e.g., the North Pacific and South Atlantic Anticyclone) of these regions. That is not to say that mesoscale coastally trapped disturbances will not be found in other areas, simply that the conditions elsewhere are not as favourable for efficient trapping and propagation of these phenomena. The subtropical location (see Fig. 1.1) of the disturbances is also favourable in that it is still an area influenced by perturbations in the upper level, synoptic westerly waves. Since these perturbations are zonal in their direction of propagation, they have a substantial component normal to the largely meridional orientation of the mountain ranges of western North America, southeastern Australia and Southern Africa. As a result, the lower level flow associated with these perturbations may be substantially blocked by the coastal mountains leading to the formation of a coastally trapped disturbance. This type of synoptic forcing of coastally trapped disturbances has been identified particularly for the coastal low of South Africa (Gill, 1977; Anh and Gill, 1981; Bannon, 1981). In general, all the coastally trapped disturbances observed to date are forced to some extent by the interaction of a synoptic perturbation with the coastal mountains. Thus, in western North America, a baroclinic trough or closed low at typically 700mb moving eastward across the coastal mountains leads to the formation of the coastally trapped disturbances there (Dorman, 1987; Mass and Albright, 1987). In southeastern Australia, a Southern Ocean cold front or migratory anticyclone approaching the southern part of the Great Dividing Range of the continent results in the formation and subsequent propagation of coastal ridges (Holland and Leslie, 1986). Although neither the North American nor the Australian studies mentioned 13 above explicitly cite the blocking of the low-level flow in the generation of the disturbances there, it is clear that this must occur in some form for these disturbances to assume the trapped form they do. Due to the common requirement of significant interaction of some synoptic feature with the coastal mountains, it is postulated here that for efficient formation of a given disturbance there must be a major component of the synoptic flow normal to these mountains. Since the coastal mountains of all three areas run largely north-south, it is clear that the mainly zonal movement of the synoptic mid-latitude and subtropical perturbations in the westerlies is favourable for the generation of coastally trapped disturbances. Another important similarity between the disturbances observed in the three regions concerns their intrinsic length and time scales (see Table 2.1). The time and length scales in this table have been obtained as follows. The Rossby radii and time scales of motion are those observed in existing studies of coastally trapped disturbances (CLW, 1984; Dorman, 1985, 1987; Holland and Leslie, 1986; Mass and Albright, 1987) while the alongshore length scale has been computed as the product of these time scales and the observed speed of propagation given by these authors. Table 2.1 shows that these scales are of the order of 100 -300 km for the internal Rossby radius (i.e., the offshore length scale), typically 1000 km or so in the alongshore direction and temporally, a day or more. Thus, the disturbances are largely alpha mesoscale phenomena in the terminology of Orlanski (1975). Although this type of scale classification is undoubtedly useful, it is felt by many that a preferable method is one based on the fundamental dynamic parameters (e.g., Emanuel, 1983). Hence, following Emanuel, it is considered that coastally trapped Table 2.1 Rossby r a d i i , alongshore l e n g t h s c a l e s , and time s c a l e s f o r the c o a s t a l l y trapped d i s t u r b a n c e s of Southern A f r i c a , A u s t r a l i a and North America. Southern A f r i c a A u s t r a l i a North America Rossby Radius 137 327 168-210 (km) Alongshore l e n g t h 3000 1150 900-1600 s c a l e of motion (km) Time s c a l e of motion (days) 6 2 3 15 disturbances are mesoscale phenomena in the sense that both background rotation and ageostrophic advection are important in their dynamics. As a result, the Rossby number Ro: Ro = U/fL (2.1) where U is a velocity scale, L a horizontal length scale, f the Coriolis parameter, and which expresses the ratio of the advective to the rotational terms, is of order unity for coastally trapped disturbances. Additionally, the time scale of mesoscale phenomena should lie between the buoyancy and inertial periods, Emanuel (1983). As will be seen later, this criterion appears only marginally satisfied, if at all, by the disturbances discussed in this thesis since the observed (Gill, 1977; Anh and Gill, 1981; Bannon, 1981; CLW, 1984; Dorman 1985, 1987; Holland and Leslie, 1986; Mass and Albright, 1987) time scale is at least 2 days and hence will be greater than the inertial period. It is therefore clear that both the numerical and dynamical definitions of mesoscale must be applied with a certain amount of caution when dealing with coastally trapped disturbances. Also of interest is the frequency of occurrence of the disturbances in the three regions. In each case, the disturbances may occur typically 4-5 times a month during the summer (CLW, 1984; Mass et al, 1986; Holland and Leslie, 1986; Dorman, 1987). With the exception of the South African case, conditions are not usually favourable for the generation of these disturbances in the winter half of the year and no observations have been reported. The reasons for the year round occurrence of the South African coastal low will be dealt with later. 16 Finally, each case is characterised by substantial land-sea temperature differences on both the meso- and the synoptic scales. These temperature differences are important in increasing the efficiency of propagation of coastally trapped disturbances by strengthening the marine inversion. This strengthening may occur directly, or indirectly via sea breezes generated by local and mesoscale temperature differences. In theory, local land breezes at night could have the opposite effect, i.e. these act to weaken the inversion. Some nocturnal erosion of the inversion has been observed during the propagation of the disturbances, for example in California (Dorman, 1985; 1987) and the PNW (Mass and Albright, 1987), but like land breezes in general, the effects were weak. On a larger scale, land-sea thermal contrast may provide additional forcing for alongshore flows. Examples of the latter would be more common on the cold current coasts such as those found along the western sides of the U.S. and Southern Africa. Thus, Chao (1985) considers that the summer intensification of the western U.S. coastal jet is due to this type of thermal forcing and, as illustrated later, warm, offshore flow originating off the interior plateau of Southern Africa is important in coastal low formation. To summarise, the above has indicated that the mesoscale coastally trapped disturbances observed propagating in the marine layers of western North America, southeastern Australia and Southern Africa exhibit a number of common features. Based on this commonality, a general theory of the disturbances is now developed. 17 2.2 Governing Equations for Coastally Trapped Disturbances Since the coastally trapped disturbances examined here satisfy Q, << N , where 0, is the intrinsic frequency of the disturbances and N is the Brunt-Vaisala frequency, and since their horizontal length scales (a Rossby radius or greater) are much greater than the scale height (about 8-10 km) of the atmosphere, the dynamics of the disturbances will be well described by the hydrostatic approximation (e.g., Gill, 1982; Pedlosky, 1987). Thus, the shallow water equations of motion (2.2 and 2.3) are appropriate. Also, the essential vertical structure of the coastal atmosphere in all three regions (e.g., Gill, 1977; Anh and Gill, 1981; Bannon, 1981; CLW, 1984; Dorman, 1985, 1987; Holland and Leslie, 1986; Mass and Albright, 1987) has been observed to consist of an inversion separating an approximately constant density cool marine layer from a deep upper layer with weak winds and stratification so a reduced gravity model is a useful first approximation. Note that in these and all other equations in this thesis, uj. = 3 u/9 t, and similarly for u x , u v , v x etc. u^ . + uu x + vuy — fv = -g'hx -f Fj. (2.2a) v t + uv x + vv y + fu = -g'hy + F 2 . (2.2b) g' = g(02- 0 1 ) / 0 2 h t + {(H + h)u}x + {(H + h)v}y = -HQ (2.2d) In (2.2), u and v are the horizontal components of the velocity, h is the displacement of the marine layer, H is the undisturbed depth of this layer, f is the Coriolis parameter, g' is the reduced gravity of the two layer atmosphere, 0^ and 0 2 are the (2.2c) 18 respective potential temperatures of the upper and lower layers, and F2 are forcing terms such as synoptic pressure gradients and Q is a source or sink rate of fluid due to thermal forcing, for example. The linear long wave speed, which is also the phase speed of a linear Kelvin wave, is c = (g'H) 1/ 2 and is about 10-20 m/s for typical values of g' and H for the coastal atmosphere in California, southeastern Australia and Southern Africa (e.g., Gill, 1977; Dorman, 1985; Holland and Leslie, 1986; Mass and Albright, 1987). Most existing models of coastally trapped disturbances (e.g., Anh and Gill, 1981; Bannon, 1981; Holland and Leslie, 1986) make the further simplification that the nonlinear advection terms are negligible. By making this simplification and by neglecting synoptic forcing and topographic variations the shallow water equations on an f plane are easily solvable analytically to yield a linear Kelvin wave or a steady gravity current as coastally trapped solutions. Despite these seemingly rather drastic restrictions, it is still advantageous to do so because it facilitates the examination of the properties of these solutions as determined by the various boundary conditions. To accomplish this, axes are introduced with the y co-ordinate measuring distance alongshore and the -x co-ordinate offshore. It is assumed that the coastal mountains may be represented by an infinitely long, vertical barrier which is a good approximation if these mountains are sufficiently steep and if their radius of curvature is significantly greater than the Rossby radius or offshore scale of the motion (Gill, 1977). Even in the most marginal case, i.e. Southern Africa where the coastal mountains describe an almost semi-circular shape, the radius of curvature of 900 km (Taljaard, 1972) is much larger than the typical Rossby radius of 100 - 200 km. The coastal mountains of western North America and southeastern Australia are, like the coastline itself, considerably less curved in orientation (compare Figs. 19 2.1, 2.2 and 2.3), hence the approximation will be even better there. Also, in all three cases the mountains typically rise quickly after a narrow coastal plain of often only 50 km or less in width. To a first approximation therefore, treating the mountains as a straight vertical barrier is reasonable. The model system is as sketched in Figure 2.4. Note that it is assumed that the height of the inversion is below that of the mountains, as is observed (Gill, 1977; Anh and Gill, 1981; Bannon, 1981; CLW, 1984; Dorman, 1985, 1987; Holland and Leslie, 1986; Mass and Albright, 1987). Since the mountains act as an impervious barrier to flow in the marine layer below the inversion, an obvious boundary condition is that the normal flow there be zero, i.e. u = 0 at x = 0. Following Gill (1982), it is appropriate to neglect u seawards to a distance corresponding to the typical offshore length scale of the motion since this scale is much smaller than the alongshore scale. In other words, the motion is assumed to be semigeostrophic and flow only occurs in the alongshore direction which is characteristic of a pure Kelvin wave (LeBlond and Mysak, 1978). This scale analysis will be performed explicitly later (see equations 2.38-2.40) but for the meanwhile it is assumed that u is negligible. It is not absolutely necessary to make this assumption, for example see Thomson (1970) for a derivation of the forced, linear Kelvin wave solution to (2.2), but it simplifies the analysis considerably and hence is used here. To reassure the reader, observations (Gill, 1977; Anh and Gill, 1981; Bannon, 1981; CLW, 1984; Dorman, 1985, 1987; Holland and Leslie, 1986; Mass and Albright, 1987) of the coastally trapped disturbances all show that this offshore velocity in the marine layer is small and that the alongshore length scale is much greater than the offshore one. With these assumptions, (2.2a-d) become 20 Figure 2.1. Topographic map of Southern Africa. The first contour inland from the coastline is the 500 m contour, that erclosing the stippled region is the 1000 m contour and that enclosing the shaded region is the 1500 m. AB = Alexander Bay, CT = Cape Town, PE = Port Elizabeth, D = Durban. On the vertical axis, the latitudes are in degrees south whereas on the horizontal, the longitudes are in degrees east. 21 Figure 2.2. Topographic map of Australia. The unshaded region refers to elevations of 0 -300m, the shaded region to 300 - 900m and the black area to topography over 900m. Adapted from Holland and Leslie (1986). 125° 22 50° h 40' 35* Sotonder Estevan PL Destruction It. * \ - L f c / ^ y : : . ^ ^ ; . . I : : : : : : : : : : : . : : ; : . Astorio 4 6 0 1 0 ' I 'M PACIFIC OCEAN *•(><>" / / | . . . . . thBend L$*< lf& O R E G M i n * &J,rm' A A B:Mm, Nor  BeCope Coif 46022 46030 Cope Mendocino ^ 46014' Pt. Areno V»^-V:^w 46013 100 km Son Francisco 46012 Monterey r» V . LEGEND i-jt '»t> 300m > 1500 m 46028 • LOCATIONS § Voncovver Island Strait of Juon o> Fuca Georgia Stroil @ Seottle.Wo. <£> Ampnitrite Pi, B C ( § Vondenberg AFB 4 6 0 1 1 * PI. Concept ion^ 46023' 125' Figure 2.3. Topographic map of western North America. The unshaded region refers to 0 - 300m, the light stippled region to 300 - 1500m and the dark shaded area to land at elevations greater than 1500 m. Adapted from Mass and Albright (1987). 23 Upper Atmosphere / x,u Mean Inversion h Height * Marine Layer e2 > > > ) s Coastal Mounta ins > ; > s s ~?—J J J J J j j Ocean x = 0 Land Figure 2.4. Schematic showing the model topography and atmospheric stratification. 24 fv= g'hx (2.3) v t = -g'hy (2.4) h t + Hv y = 0 (2.5) The general solution (Gill, 1982) to (2.3-2.5) is : h = e' x / R F(y + ct) + e x / R G(y - ct) (2.6) v =-(g7H)1/2{e-x/R F(y + ct) - e x / R G(y - ct)} (2.7) where R is the internal Rossby radius defined as : R = (g'H/f2)1/2 (2.8) Two types of boundary conditions are now applied to determine possible motions. These boundary conditions will yield two different solutions; namely, a non-dispersive, linear Kelvin wave and a steady, linear, coastally trapped gravity current. Firstly, assuming that the inversion layer displacement and alongshore velocity v vanish far from the coast leads to the Kelvin wave solution of the form h = e x / R G(y - ct) ' (2.9a) v = (g ' /H) 1 / 2 ^/ 1 1 G(y - ct) (2.9b) Note that the internal Rossby radius is the e-folding cross-shore scale measuring the trapping of the Kelvin wave to the coast. Also shown by these equations, and applied by Gill (1977), Maxworthy (1983), Dorman (1985) and Holland and Leslie (1986), is the fact that the alongshore structure G(y - ct) of the motion is quite general in the 25 linear case. Thus, any arbitrary function G that travels-in the +y direction at a phase speed c = (g'H)-'-/^  is allowed. Examples would be an oscillatory linear wave, a pulse or a more complex wave shape. Secondly, a steady coastally trapped gravity current solution may be derived from (2.6-2.7) by assuming the existence, at a constant distance L' offshore, of a density front that separates the cool, denser fluid of the gravity current from the ambient atmosphere. At the density front L', the inversion displacement is assumed zero. Since a steady solution is required, the functions F and G must now be constants so that (2.6-2.7) yield : • h = { e x / R . e-(x + 2L')/R}G (2.10a) v = -(g'/H)1/2{e-(x + 2L'/R) . e x / R } G ( 2 1 0 b ) The constant G may be specified in terms of a known steady velocity V Q of the gravity current at the coast x = 0 as G = -(rI/g')1/2V0/(e"2L /R -1). This velocity is the steady rate of spread of the buoyant lower layer alongshore under gravity - hence the term gravity current. For a two layer fluid, V 0 is given by the long wave speed (g'H) 1/ 2, where H is now the interface depth at the coast, multiplied by a constant factor k (Benjamin, 1968; Griffiths, 1986; Smith and Reeder, 1988). As discussed in more detail in Griffiths (1986), equations (2.10a-b) show that there are two independent trapping scales for this gravity current solution, one based on the internal Rossby radius and the other on the position of the density front. In practical terms, it is usually difficult in atmospheric (as opposed to oceanic or laboratory) applications to identify the exact frontal position and so in accordance with the observations of coastally trapped disturbances the two scales are regarded as 26 being equivalent (e:g., Baines, 1980; Dorman, 1987; Mass and Albright, 1987). Also, , note that the offshore displacement and velocity profiles of the gravity current are different from the simple exponential decay away from the coast of a Kelvin wave (cf. 2.9a-b), a fact that will be important later. From the above discussion, it is clear that the simple linear shallow water equations for a rotating, stratified and flat-bottomed fluid bounded by a side wall allow at least two types of topographically trapped solutions. These solutions are a linear, non-dispersive Kelvin wave propagating with phase speed (g'H)!/2 and a steady coastal gravity current of speed V Q given by k(g'H)1/2. The factor k was shown by Benjamin (1968), using Bernoulli's principle, to take the value -y/2 for a steady gravity current in a deep, non-rotating and inviscid fluid. Using a similar argument for the rotating case, Griffiths (1986) showed that k > y/2. However, Smith and Reeder (1988) in a comprehensive study of the validity of gravity current models to atmospheric fronts, state that, on the basis of empirical studies, k is best taken as unity. The discrepancy between this value for k and y/2 is presumably to account for friction, which would reduce the flow speed and hence cause the flow to be unsteady, and the dependence of k on the depth of the gravity current flow to that of the total fluid depth. Both of these effects were ignored by Benjamin (1968) and Griffiths (1986); however, the former was treated by Kubokawa and Hanawa (1984) using a model that included the nonlinear advection terms and a shock wave solution for the gravity current head where frictional effects are important. Some uncertainty also exists as to whether the gravity current depth should be taken as that of the gravity current head or of the main flow behind the head (Smith and Reeder, 1988). To avoid this uncertainty, it is preferable 27 (Wakimoto, 1982; Seitter and Muench, 1985; Garratt and Physick, 1986; Mass and Albright, 1987; Smith and Reeder, 1988) to determine the gravity current speed from the observed surface pressure difference between the ambient fluid immediately ahead of the gravity current head and the head itself as c = 0.79(8P/ p)1/2 (2.11) where 5P is this pressure difference and p is the mean density of the ambient fluid. This formula has been shown by these authors to give an accurate prediction for the observed speed of a variety of atmospheric gravity currents provided that the surface pressure difference is measured with sufficient resolution (hourly). In general, despite the variation in gravity current speeds predicted by the available theories, it is known empirically (Griffiths and Hopfinger, 1983; Maxworthy, 1983; Kubokawa and Hanawa, 1984) that this speed is somewhat different to that of a Kelvin wave, a fact which will be important in determining whether or not the observed coastally trapped disturbances in North America (Mass and Albright, 1987) and Australia (Holland and Leslie, 1986) are in fact Kelvin waves. This use of the different propagation speeds to distinguish between coastal gravity currents and Kelvin waves provides a simple but, unfortunately, not critical test due to the noted uncertainties in the gravity current theory and because, in some cases, the data may be of insufficient resolution. The other difference between the Kelvin wave and coastal gravity current solutions that has emerged from this simple linear analysis concerns the offshore structure of the two phenomena. This difference results mathematically from the imposition of the boundary conditions and reflects the fundamental physical distinction between 28 gravity currents, which consist of a flow of different density to that of the ambient fluid into which the gravity current is intruding, and Kelvin waves, which propagate on the interface between two fluid layers of unequal density. In the next section, this physical distinction between gravity currents and Kelvin waves and its consequences for the marine layer are explored in greater detail. Emphasis is placed upon the importance of the scale of the external forcing in determining whether gravity current or Kelvin wave motion occurs. 2.3 Differences between coastally trapped gravity currents and Kelvin waves An immediate consequence of the fundamental distinction between the two types of motion is that a gravity current involves a transfer of fluid mass whereas a Kelvin wave is only able to transfer information and energy between points in the fluid. Thus, for a gravity current to be able to flow any great distance along the coast a large reservoir of fluid of different density (and hence temperature) is required to continually feed the gravity current intrusion. Put another way, if a rotating, stratified fluid with a side wall is externally forced, then a coastal gravity current will only result if the forcing is sufficiently large or of long enough duration to provide extended input of different density fluid. According to Griffiths (1986), large means comparable to or greater than a Rossby radius while long may be taken as similar to or longer than an inertial period. On the other hand, if the forcing does not meet these requirements and is more impulsive in nature, then the resulting motion is likely to take a Kelvin wave form. In the context of the coastal atmosphere, this reservoir must take, the form of a large area of raised marine layer containing a cool, dense airmass. Topographic trapping of 29 the reservoir against the coastal mountains should also occur so that a sustained gravity current flow may take place without dissipation or destruction of the cold air source by the background synoptic flow. Hence, large bays or bights are preferential areas for the generation of coastally trapped gravity currents, witness the formation of the North American disturbances in the California Bight (Dorman, 1985, 1987) and the Australian ones in the Great Australian Bight (Holland and Leslie, 1986). Since these gravity current events require a continual influx of denser air, their propagation results in a long term (several days to a week) displacement of the marine layer and corresponding drops in temperature and changes in the windfield. Thus, the marine layer all the way from the gravity current nose to the cold air source should be displaced throughout the gravity current event and once this is over the entire length of this marine layer should return to its pre-disturbance condition at about the same time. Due to a substantial reservoir being necessary, the generation mechanism for gravity currents must either involve a continual input of different density fluid into the reservoir or else sufficiently large synoptic forcing to allow enough of this fluid to collect for outflow over the 3-5 days observed for the North American and Australian disturbances. In the coastal ocean, the former type of generation could involve estuarine outflow, or icepack melt but in the coastal atmosphere the only conceivable continual input mechanism might be valley outflow from a continental airmass blocked otherwise from the coast by mountains. Such a mechanism may be responsible for some of the strong gap winds observed during winter down fiords in the PNW (Overland and Walter, 1981) but since the synoptic flow in the cases of interest is along- or onshore it is not an appropriate mechanism here. Thus, some type of synoptic forcing is necessary for gravity current generation on the scale 30 observed in the cases of interest here. Consistent with this argument, coastally trapped disturbances of gravity current form observed in western North America have, been generated by a large, baroclinic trough in the westerlies (Dorman, 1987; Mass and Albright, 1987) while th ose in southeastern Australia result after the approach of a large anticyclone towards the mountains there (Holland and Leslie, 1986). These types of forcing fall within the scope of the generation of inertial coastal gravity currents via the broad (greater than a Rossby radius) geostrophic flow onto the coastal boundary stated by Griffiths (1986) in his review of gravity currents in rotating fluids. Other characteristics typical of gravity currents are unsteady, often spurting flow (Simpson, 1987) and the existence of a density front between the gravity current and the ambient fluid at which the interface displacement will be zero. Fig. 2.5a illustrates this density front for a typicalcoastally trapped gravity current in the Californian marine layer - the curved nature of the front such that the leading edge at the coastal mountains progresses ahead of that at the seaward boundary is always observed (Dorman, 1987; Mass and Albright, 1987). Fig. 2.5b shows a side view of a typical (not necessarily coastally trapped) atmospheric gravity current where a density front would exist between the cold gravity current air and the warmer ambient atmosphere. Since the unsteady gravity current flow is generally unstable to both the Kelvin-Helmholtz and the baroclinic mechanism (Griffiths, 1986), various types of waves are generated. These waves may form either at the coast or offshore at the density front that separates the current from the ambient fluid (Stern, 1980; Stern et al., 1982; Kubokawa and Hanawa, 1984; Griffiths, 1986; Paldor, 1988). Figure 2.5a. Perspective view of a coastally trapped gravity current in the Californian marine layer showing the position of the curved density front x = -L'(y,t). Note that the displacement of the gravity current is maximum near the coastal mountains and is zero at the density front, i.e., at the front the marine layer depth equals its mean value of 400 m. Adapted from Dorman (1987). Figure 2.5b. Side view of a typical atmospheric gravity current showing the greater depth of the head than the feeder flow behind. Adapted from Seitter and Muench (1985). 32 The existence of a density front between the gravity current and the ambient fluid suggests that as a coastal gravity current moves alongshore beneath the inversion, abrupt (i.e., minutes) changes in the density and hence temperature should be observed in the marine layer. Such abrupt changes have been recorded during the July, 1982 (Dorman, 1987) and May, 1985 (Mass and Albright, 1987) coastal gravity currents along the west coast of North America but were not reported during the May, 1982 event (Dorman, 1985; Mass and Albright, 1987) which reinforces the solitary Kelvin wave interpretation of this event (Dorman, 1985, 1988). Kelvin waves, on the other hand, do not necessarily require the existence of a reservoir of different density fluid. Instead, they are a type of gravity wave that can only exist at the vertical side boundary of a rotating fluid and are only influenced by rotation in terms of the e-folding trapping scale (Rossby radius) of the alongshore velocity and displacement to this boundary. Propagation of a Kelvin wave event should be steadily progressive and not spurting back and forth along the side boundary as may occur for a coastally trapped gravity current (Stern et al., 1982; Simpson, 1987). Additionally, there is no density front between the propagating Kelvin wave and the ambient fluid so that changes in the density and temperature of the fluid should not occur abruptly during propagation as they do for a gravity current. If, however, nonlinear terms are sufficiently strong then resulting wave steepening may result in more abrupt changes. Counteracting the steepening are the effects of dispersion, which when a balance occurs, give rise to a solitary Kelvin wave for which one also does not expect abrupt changes in the density or temperature with propagation. Hence, in general, the marine inversion and the surface pressure, temperature and wind fields should be displaced in an pulse-like fashion as a Kelvin wave disturbance propagates past a given station. Once the wave crest/trough has 33 passed this station, conditions in the marine layer should return to their initial pre-disturbance state quickly and not remain displaced as in a gravity current event. Forcing mechanisms for Kelvin waves are. various. In the coastal ocean, direct wind stress is probably the most efficient and common generator (Thomson, 1970; LeBlond and Mysak, 1978; Gill, 1982). Also, atmospheric pressure oscillations due to storms (LeBlond and Mysak, 1978) and buoyancy advection associated with river outflow or icemelt (Ikeda, 1984) may generate coastal ocean Kelvin waves. Note that all these mechanisms have atmospheric analogues. Interaction (involving some type of blocking) of the incident synoptic flow with the coastal mountains may generate Kelvin waves in the coastal atmosphere (Gill, 1977; Anh and Gill, 1981; Bannon, 1981). Like most of the other generation mechanisms already mentioned, blocking of the synoptic flow could also generate a gravity current disturbance in the marine layer. Thus, as implied earlier, it is not the type but rather the scale of the forcing that determines whether or not a gravity current is generated. To re-emphasize, this forcing must be large or last long enough to allow reservoir formation and hence the continual input of denser fluid required for gravity current flow. For ease of reference, the various distinctions between gravity currents and Kelvin waves discussed above are now summarised. Kelvin Waves Transport Energy, information Gravity Currents Mass, denser fluid Offshore Exponential decay, structure no density front Density front Motion Propagate on the inversion. Steadily progressive "Smooth", wavelike changes in fields with oscillatory return to initial state. Speed = (g'H) 1/ 2 Flow in the marine layer. Unsteady, spurting Abrupt changes in wind, temperature fields with long term displacement and irregular return to initial state. Speed = k(g'H) 1/ 2 Forcing - Localised, impulsive No reservoir Large scale and long term Reservoir of different density fluid. In terms of identifying a coastally trapped disturbance as a Kelvin wave or a gravity current in the marine layer, it is considered that the most practical methods depend on matching the observed propagation speeds with those predicted theoretically; determining whether the displacement of the inversion and marine layer conditions is unsteady or steadily progressive; and the use of satellite imagery and sounding data 35 in the formation region to see whether the large reservoir necessary for gravity current propagation has formed. 2.4. Some forced solutions of the shallow water equations Since the above discussion has indicated the importance of the forcing, the governing equations (2.2a-d) are considered for various special forms of the forcing terms F^, F 2 and HQ. Firstly, the effects of a source or sink of fluid mass are considered so that (2.2a-d) become : uj. + uu x + vuy — fv = —g'Hx (2.12a) v t + uv x + vv y + fu = -g 'h y (2.12b) h t - (uh)x + (vh)y .= w s , • • (2.12c) where the substitution h'• = h + H has been made and w g is a vertical velocity that arises from the synoptic forcing. If w g is negative (positive) then it is a subsidence (uplift) and acts to transport fluid into (out of) the marine layer. Cross-differentiation of (2.12a-b) and substitution for the horizontal divergence from (2.12c) leads to the vorticity equation : • D / D t ( v x - u y + f) = - h - 1 ( v x - u y + f)(w s-,Dh/Dt) which on dividing by (v x -u y + f) can be rewritten as D / D t { l n | v x - u y + f|} = -E- 1 (w s -DE/Dt) From this equation it is clear that the contribution of a subsiding wind (wg negative) to the marine layer vorticity budget is to yield an increase with time of the absolute 36 value of the absolute vorticity |v x - u v + f| which for an f plane in either Hemisphere means the production of cyclonic relative vorticity. Similarly, an ascending wind will lead to the production of anticyclonic relative vorticity. Now, synoptic forcing which causes low level onshore flow will lead to convergence at the coastal mountains and hence uplift whereas offshore flow will be associated with divergence at the mountains and subsidence. From the above vorticity argument, it is evident that offshore (onshore) flow will tend to cause a mesoscale low (high) to form in the marine layer. Motivation for considering this situation is that the North American and Australian cases are all coastal highs (Dorman, 1985; 1987; Holland and Leslie, 1986; Mass and Albright, 1987) whereas the Southern African events are all coastal lows (Gill, 1977; CLW, 1984). If the equations may be linearised, i.e., the motion has small Rossby number and particle speeds small compared to the wave speed c, then an analytic solution may be obtained. Linearising (2.12a-c) and eliminating the horizontal velocities leads to an equation for w = E^ alone : (-f2R2v 2 + a t t + f 2 ) z = ( + f 2 ) W g ( 2 1 3 ) whereV 2 is the horizontal Laplacian, 3^'= 3 ^/ Bt 2 and R is the internal Rossby radius. The boundary condition of no normal flow at the coast implies: wxt + f w y = 0 at x = 0 (2-14) Although it is possible through the use of Fourier and Laplace transform techniques to obtain a general analytic solution to (2.13-2.14). (see, Thomson (1970)), it is more 37 intructive (e.g., Ikeda (1984)) to consider the equations under the sub-inertial, long wave conditions typical of coastally trapped disturbances. Thus, the fact that the alongshore length scale is much greater than the across-shore scale (Table 2.1) is used to replace the Laplacian by 3 ^  and since the typical time scale of the disturbances is 2 days or more (Table 2.1) the second order time derivative in (2.13) can be neglected in comparison with the f2 term so that : wxx ' R " 2 w = -R"2 ws (2-1 5) The solution is : z =A(y,t)e x/R-(2R)- 1{e x/RJe- x/Rw sdx-e'^Je^Rwgdx} (2.16) 0 -oo after using the condition that far from the coast the flow should vanish. To determine the time dependent along-shore structure A(y,t), equation (2.16) is substituted into the boundary condition (2.14) : 0 .At + fRA y -. |e x / R {(w s ) x t + f(ws)y}dx = 0 (2.17) -oo From the method of characteristics (e.g., Whitham, 1974) (2.17) is seen to represent a forced wave with phase speed fR = (g'H)-'-/2 propagating along the right-hand (left-hand) coast in the Northern (Southern) Hemisphere and thus the solution (2.16) describes a Kelvin wave trapped to within a Rossby radius of the coast. Note that 38 this form of solution, i.e., a Kelvin wave, is exactly that found in the more general model of Thomson (1970) mentioned above. Now consider the effects of the sign of wg. For wg negative, i.e., subsiding, (2.16) indicates the contribution of the forcing is to cause a relative decrease in z and hence a relative decrease with time of the inversion displacement. In other words, a lowering of the inversion height occurs under subsiding wind conditions. This lowering of the inversion height together with the production of cyclonic vorticity shown above indicates that the subsidence wind forcing results in coastal low rather than coastal high formation. If the forcing leads to wg positive (i.e., uplift) then (2.16) shows that a coastal high will form. Having shown in very simple terms how a subsidence/uplift velocity may lead to forced Kelvin wave propagation, the influence of periodic variations in this forcing are considered. Motivation for considering periodic forcing follows from the observation of a distinct 6 day cycle in the surface pressure spectra measured at coastal South African sites (Preston-Why te and Tyson, 1973; Kamstra, 1987) and the identification of this cycle with coastal low dynamics (Gill, 1977; Bannon, 1981). Thus, the subsidence velocity is expanded into a Fourier series, one component of which is written as W g = W(x)exp{i(ky - Ot + e)} ' (2.18) where k is the wavenumber, e is the phase lag between the forcing and the disturbance and Q is the frequency corresponding to a period of 6 days. For simplicity, consider a subsidence velocity that is constant inshore and zero offshore, i.e., 39 W(x) = W Q , a constant -a < x < 0 (2.19) 0 x < -a where a is given the value 50 km. Hence, the inversion displacement has the form h = h(x)exp(i(ky - fit)) (2.20) so that (2.13) and (2.14) become h x x - R - 2 h = -W*R- 2e i( 1 t/ 2'-*) (2.21) ch x + fh = 0, at x = 0 (2.22) where c = n/k = (g'H) 1/ 2 ; w* = wQ(i - n2/i 2)/n , (2.23) . The solution to equation (2.21) is h =H G e x / R - W * ^ / 2 + e){e x/ R - e-a/Rsinh(x/R) - 1} ; -a < x < 0 (2.24) and h ={H0 - W V W 2 + t)(i . cosh(a/R))}ex/R ; x <-a (2.25) where H Q is the inversion height at the coast. Note that the boundary condition at the coast (2.22), which was used previously to determine the alongshore structure of (2.16), has been made redundant by the assumption (2.20). Also, the existence of a simple Kelvin wave solution is not dependent on the forcing being periodic in either space or time. As shown in Ikeda (1984), a fixed subsidence velocity gives rise to a propagating Kelvin wave solution in the far field and a stationary baroclinic eddy centred on the location of the forcing. In general, the effect of the periodic forcing 40 near the coast will be to modify the offshore structure of the Kelvin wave from the simple form (2.9a-b). It is shown now that inclusion of the nonlinear alongshore advection terms in the equations lead to a further modification of the offshore structure. As before, the equations are simplified by the assumption that the across-shore flow u is negligible so that (2.12a-c) become fv = g'hx • (2.26a) v t + vv y + g'hy = 0 (2.26b) h t + Ev y + vh y = ws (2.26c) Substituting (2.16a) into (2.26c) yields a forced equation in the displacement h: h t + gT^hh^y =w s (2.27) To simplify matters, the following scaling is introduced. The inversion depth h is scaled by its maximum value at the coast H, the offshore displacement by the Rossby radius R, the alongshore scale is taken as the reciprocal of the wavenumber k, and time is scaled by the reciprocal of the frequency Q. Hence (2.27) becomes (where variables are now non-dimensional) *H + (hh x) y =w s/ftH (2.28) As previously, it is assumed that both h and wg are wavelike (i.e., 2.18-2.20) and a co-ordinate system (2.29) moving with the wave in the inversion displacement can be introduced. Y = y - t (2.29) 41 Substituting and integrating once (the integration constant is formally assumed to be zero corresponding to zero initial displacement) yields Y nn x -h'= (fiH)"1 [w sdY (2.30) y and hence Y h x = 1 + (Efffl)-1 [ ws dY (2.31) Equation (2.31) describes the offshore slope of the inversion depth (scaled) when nonlinearities are included and will be compared with the slope obtained for the linear case. Then, the linear equivalent of (2.28) is: h t + h x y = w s/fiH (2.32) and, after introducing co-ordinate system (2.29) and integrating as before, this becomes: Y h x = h{l + (hftH)-1 wsdY} (2.33) y Equation (2.33) is the slope of the inversion depth (scaled) for the linear case. Now, since this scaled depth must lie between zero and unity, comparison of (2.33) and (2.31) shows that the decrease of the inversion depth offshore proceeds at a greater rate for the nonlinear than for the linear case. In other words, when nonlinearities are included, the decay scale is no longer the Rossby radius of the linear case but some' fraction thereof. This result is consistent with a decay scale of a half Rossby radius 42 observed in the rotating tank experiments for strongly nonlinear Kelvin waves by Maxworthy (1983) and Griffiths and Hopfinger (1983) and with the theory of Grimshaw (1985). The latter author showed that for weakly nonlinear Kelvin waves, the decay scale was modified to include terms fractionally proportional to the Rossby radius. Returning to the alongshore behaviour, note that (2.26b-c) can be put in the characteristic form (Whitham, 1974) Y t + A(V)V y = w (2.34) where V and W are column vectors containing components h,v and wg,0 respectively and A is the 2x2 matrix (2.35). A(Y) = [v E (2.35) In the case of W being zero, which is true for distances offshore of a, the system (2.34) is hyperbolic and has characteristic velocities v ±(g'h)-'-/^ Whitham (1974). Then, a simple wave moving eastwards into a reduced gravity atmosphere with lower layer of depth H' is given by E = H(Y), v = 2(g'H)1/2 . 2(g'H')1/2 on y = Y + {3(g'H(Y))1/2 - 2(g'H')1/2}t (2.36) (2.37) 43 Note that the solution (2.36 - 2.37) requires the lower layer of the atmosphere ahead of the wave to be still. Although this requirement seems restrictive, particularly in view of the fact that there are usually climatological winds blowing ahead bf the disturbances (e.g., CLW, 1984; Dorman, 1985), the lower layer immediately ahead of the wavefront can be considered still in the sense that there are no other meso- or synoptic scale disturbances in the neighbourhood (within several Rossby radii say). Now, whether or not the wave will steepen and eventually break depends on whether it is a wave of elevation or depression. For a wave of depression (e.g., the Southern African coastal low) in which the inversion layer is lowered beneath its undisturbed value, steepening and potentially, breaking, may occur at the trailing edge of the low but not at the wave front (Stoker, 1957; Whitham, 1974). The reason steepening does not occur at the wave front is that the slopes of the straight line characteristics, given by the term inside parentheses in (2.37), can be shown to decrease as time increases (Stoker, 1957). Hence, the family of straight characteristics diverges on moving away from the time axis. On the other hand, for waves of elevation (e.g., the N. American and Australian coastal ridges) the characteristics may be shown to converge (Stoker, 1957; Whitham, 1974) so that intersection and a multi-valued solution eventually occur, which in turn leads to either breaking or bore formation at the leading edge. The reverse argument shows that nonlinear steepening may occur at the trailing edge of a wave of depression. For the case W non-zero, where W is now a function of y and t, it can still be shown that system (2.34) allows simple wave solutions with characteristic velocities y ± ( g ' h ) 1 / 2 ) stoker (1957) and hence, that breaking or bore formation may occur only at the leading (trailing) edge for a wave of elevation (depression). 44 2.5. Nonlinear semigeostrophic theory of coastally trapped disturbances In terms of the linear, unforced equations (2.3-2.5), it is solely the boundary conditions which determine whether Kelvin wave or gravity current motion occurs. However, as will be seen below, inclusion of the nonlinear advection terms indicates that the dynamics of coastal gravity currents and Kelvin waves are related so that an initial gravity current motion may evolve into nonlinear and eventually, solitary Kelvin wave disturbances. This evolution has been observed in the atmosphere for small scale, non-rotating flows on the nocturnal inversion in northern Australia (Christie et al., 1978, 1979) and in rotating tank experiments (Maxworthy, 1983; Kubokawa and Hanawa, 1984; Simpson, 1987). It is argued in herein that a similar situation occurs for the larger scale, rotation influenced coastally trapped disturbances of western North America and southeastern Australia. What follows is an adaptation of the theory of Kubokawa and Hanawa (1984) to include forcing and appropriate time and length scales for coastally trapped disturbances in the atmosphere. The influence of dissipative losses on the motion will be considered later. Although the Kubokawa and Hanawa theory was developed for the oceanic case, the fundamentals apply to any rotating, stratified fluid (e.g., the coastal atmosphere) and so only those details which differ from their study will be discussed at length here. In order to see the relationships between a coastal gravity current and nonlinear Kelvin and frontal waves in a rotating, stratified fluid with a side boundary, a multiple scale analysis is applied to the shallow water equations (2.2a-d). Because many of the observations of coastally trapped disturbances appear to form initially as a coastally trapped gravity current which then evolves into solitary wave like features, it is felt that the reductive perturbation theory of Kubokawa and Hanawa (1984) in which this type of evolution is explicitly analysed is relevant. It should be mentioned here that an alternative theory (Stern, 1980; Stern et al., 1982) of the problem studied by Kubokawa and Hanawa (1984) is not valid in this context because of the assumptions by Stern and his co-workers of zero potential vorticity (Kubokawa and Hanawa assume constant potential vorticity) and uniform initial conditions that govern the downstream evolution of the gravity current. The former assumption amounts to the initial angular momentum of the coastal flow being zero and assumes that the buoyant intrusion of the different density fluid emerges from a continuous point source. While this may be appropriate for flows produced in the coastal ocean from estuarine sources, it is not valid for coastally trapped disturbances in the atmosphere generated by synoptic forcing. The second assumption may be violated because, in many cases, the gravity current generates upstream propagating waves which modify the initial conditions (Kubokawa and Hanawa, 1984). Following Kubokawa and Hanawa, it is appropriate to scale the nonlinear shallow water equations (2.2a-d) as follows. Time is scaled by the reciprocal of the Coriolis parameter f multiplied by a small parameter d, the alongshore displacement by the Rossby radius R divided by d, the across-shore displacement by R, the alongshore velocity by the long gravity wave speed fR, the across-shore velocity by this speed multiplied by d and the inversion displacement by the typical inversion height H. Forcing, in the form of an external synoptic pressure gradient, is similarly scaled (e.g., Gill, 1977) so that equations (2.2a-d) become : ;(ut + uu x + vuy) - v = - h x - P x v t + u v x + v v y .+ u = • h y • P y h t + uh x + vh y + (1 + h)(ux + v y) = 0 (2.38) (2.39) (2.40) The parameter d is defined as : d = RH/La (2.41) where L is an alongshore length scale and a is a typical amplitude of the disturbance. This definition of d differs from the R/L used by Kubokawa and Hanawa (1984) and has been introduced so as to have a direct measure of the nonlinearity in the analysis. In addition, the analysis makes use of the conservation of potential vorticity for an inviscid, homogeneous shallow water layer. Cross-differentiating (2.38) and (2.39) followed by subtraction and then substitution from (2.40) leads to an equation expressing this conservation. Assuming the initial potential vorticity (in dimensional units) to be f /H 0 , where H 0 is the initial value of the inversion height 1 leads to the where the constant k = H 0 / H and the substitution R = 1 + h has been made to give exactly the same form as Kubokawa and Hanawa. Note that h identically satisfies (2.38 - 2.40). form : v. • x - d2 u y + 1 = k 2 E (2.42) Following Kubokawa and Hanawa, the solutions of (2.38-2.40) are then separated into semigeostrophic (subscripted s) and ageostrophic (subscripted a) parts, where the former will be an exact solution in the limit d 2 -»0: 47 U = Ug + ^ ^ v = v s + d 2 v a (2.43) h = h s + d 2 h a As previously, the boundary conditions of zero interfacial displacement at the density front x = -L' (see Fig. 2.5a for location of this front in relation to the gravity current) and zero across-shore velocity at the coast are applied. Now, however, it is no longer assumed that the frontal position or the alongshore velocity at the coast are steady (as in Section 2.3); instead Kubokawa and Hanawa (1984) allow the gravity current to be unsteady and consider specifically the alongshore velocity V(y,t) at the density front x = -L'(y,t), i.e., both V and L' are now functions of time and alongshore displacement. To investigate the possible forms of V(y,t), Kubokawa and Hanawa (1984) introduce the condition that the across-shore velocity at the density front x = -L'(y,t) is given by the convective derivative of this frontal position. This boundary condition follows from the fact that the density front acts as a free streamline so that the fluid elements on this streamline remain there for all times. Such a streamline then behaves as a material boundary separating gravity current fluid from ambient fluid with no exchange between the two. Hence the appropriate boundary conditions are : At the coast x =. 0 : u = 0 At the front x = -L'(y,t) : u = DL/Dt, h = 0 (2.44) (2.45) Setting d 2 , and for convenience the synoptic forcing P both to zero i n (2.38) and (2.43) then yields the semigeostrophic solution v s = -k_1sinh k(L'4 x) + V(y,t)cosh k(L'+ x) (2.46) h s = k"2{l - cosh k(L'+x) + kV(y,t)sinh k(L'+x)} (2.47) where V(y,t) is the alongshore velocity at the density front x =-L'(y,t). Upon substituting (2.46-2.47) back into (2.39) and evaluating the resulting equation at x = 0 and -L', the following nonlinear system is obtained Y t + A V y = 0 (2.48) where.V is a vector containing components (V,L') and A is a 2x2 matrix (equations A-l to A-4 of Kubokawa and Hanawa, 1984) that is a function of V. Further details are given in Kubokawa and Hanawa (1984); the modification to their analysis is the re-definition of d. Note that including non-zero synoptic forcing in the analysis will lead to an equation of the form (2.49): Yt + B V y = - P y where P is a vector containing this forcing evaluated at x = 0, -L' and where the matrix B will have slightly different components to those of A that will depend on the modification to (2.46-2.47) by the forcing. Neither modification alters the evolution of the solitary wave derived below because this wave is an asymptotic solution and is therefore essentially independent of precisely how it is formed (Grimshaw, 1988 pers. commun.). Hence, the only effect of non-zero external forcing is to modify the exact displacement and velocity profiles of the initial gravity current disturbance given in (2.46-2.47). (2.49) 49 Using standard techniques (see for example Whitham (1974)), (2.48) is multiplied by the left eigenvector 1 of the matrix A so that the characteristic form (2.50) results ldV = 0 on dy = c (2.50) dt dt ]A=]c If external forcing is included then (2.50) becomes 1* dV + P v = 0 on dy = c (2.51) ." dt y dt where 1 is the left eigenvector of matrix B and 1 B = 1 c. Equation (2.50/2.51) is a hyperbolic system which represents two independent waves with phase speeds given by the two characteristic velocities or eigenvalues c of the 2x2 matrix A or B (Whitham, 1974). As shown by Kubokawa and Hanawa (1984), one of these waves represents a Kelvin wave which is trapped against the coast and which has a phase speed greater than V (i.e., this wave is advected by the gravity current), while the other is trapped against the density front and because it has a phase speed less than V, propagates upstream relative to the current. Since these two waves are nonlinear, steepening of the wavefront must occur. Opposing this steepening are dispersive or spreading out effects which result from the action of the mean gravity current. To determine what happens when these effects balance, Kubokawa and Hanawa expand the variables in (2.43) in terms of the perturbation parameter d 2 and equate the nonlinear tendency (at order d^ ) with the dispersive (see Appendix 1 for details including dissipative effects). Introducing a slow time scale given by the original scale 1 /df multiplied by d ,^ Kubokawa and Hanawa show that a balance exists on this scale between these two tendencies as described by the KdV equation for the amplitude A of the Kelvin and frontal waves: A T •+ nAA T + m A T T T = 0 (2.52) where m and n are constants measuring the relative strengths of the nonlinearity and dispersion, the phase variable T = d2(y - fRt) with y measured alongshore and T is the slow time scale : T = d' 5 f 1 (2.53) For the reduced gravity model used here, m is 0.1 and n is 5 (Kubokawa and Hanawa, 1984) but in the continuously stratified case these coefficients depend directly on the stratification (see Gear and Grimshaw (1983) and Grimshaw (1985) for details). The solitary wave solution to (2.52) is , A(T) = a sech2(-r/Ls) (2.54) where the constant a is the amplitude of the solitary wave and the effective wavelength of the wave L s is : L s = (12m/na)1/2 (2.55) To summarise, the gravity current evolves solitary Kelvin (downstream propagating) and frontal (upstream propagating) waves with alongshore structure (2.54) on the time scale (2.53) which are trapped with across-shore structure e x / R to the coast and the density front respectively. Thus, the solitary Kelvin wave displacement is: h(x,y,t) = aex/Rsech2{d2(y - £Rt)/L s} (2.56) with phase speed: c = {fR(l + a)}1/2 = ( g'H(l + a)} 1/ 2 (2.57) It must be emphasized that for the solitary waves to evolve, dispersion must balance the nonlinear steepening. This dispersion occurs through the action of the mean alongshore gravity current and will therefore be weak at and ahead of the leading edge of the current (Kubokawa and Hanawa, 1984). Thus, if there is significant across-shore flow at this leading edge then there will be no dispersion and, instead of a solitary Kelvin wave forming, nonlinear steepening will cause a shock wave to develop on the time scale (2.53) (Kubokawa and Hanawa, 1984) with speed: c = l ^ g ' H J 1 / 2 (2.58) where H u is not the mean inversion height H but the depth of the gravity current flow behind the leading head. Equation (2.58) was derived by assuming conservation of mass and momentum across the shock (Kubokawa and Hanawa (1984), a more realistic assumption than that of conserved potential vorticity used by Nof (1986) in his theory of geostrophic shock waves. A quantitative indication of the likelihood of shock as opposed to solitary wave formation may be obtained by considering the parameters a/H and H/L (L being the 52 alongshore length scale) measuring nonlinearity and dispersion respectively. By analogy with surface gravity waves, e.g., Lighthill (1958, 1978), as a/H approaches unity, steepening and shock formation take place. For a/H and H/L both small, linear sinusoidal shallow water waves occur. In between these extremes is the solitary wave in which the dispersive effects balance the nonlinearities. It is considered that the weakly nonlinear theory of Kubokawa and Hanawa adapted here offers a reasonable first approximation to the evolution of an initial gravity current flow into a propagating solitary Kelvin wave (dispersion important) or shock wave at the leading edge (dispersion negligible). Rotating tank experiments and the fully nonlinear numerical simulations of Holland and Leslie (1986) and Hermann et al (1989) have indicated that the evolution process is likely to be strongly nonlinear so that one should not expect more than order of magnitude accuracies from theories like Kubokawa and Hanawa (e.g., Grimshaw, 1985; 1988 pers. commun.). In all the events examined herein, application of the Kubokawa and Hanawa theory gave considerably better than order of magnitude accuracy. In summary, the modified Kubokawa and Hanawa (1984) theory offers a convenient theoretical description of coastally trapped disturbances which allows for the existence of both gravity currents and Kelvin waves. Thus, the viewpoint of Holland and Leslie (1986) that the Australian coastal ridging forms initially as a coastal gravity current which then evolves into a Kelvin wave disturbance that propagates through and ahead of the current is theoretically justified by the analysis given here. As discussed later, the disturbances observed in the Californian and PNW marine layers by Mass and Albright (1987) and Dorman (1987) may be more fruitfully interpreted in a similar way, namely as topographically trapped gravity currents with associated evolving nonlinear Kelvin waves. 53 Note that the Southern African coastal lows appear to be well described in most cases by linear Kelvin wave dynamics (e.g., Anh and Gill, 1981; Bannon, 1981; CLW, 1984 and see below) and so would fall outside the realm of the Kubokawa and Hanawa theory. Since this theory assumes nonlinearities are important, the next step in the analysis is to investigate the validity of this assumption by determining the typical values of the parameters that measure the contributions of the various terms in the momentum balance during the formation of coastally trapped disturbances. 2.6 Scale Parameters characterising the Dynamics of Coastally Trapped Disturbances a). Formation A formal scale analysis and a subsequent numerical solution of the primitive equations describing the flow of a Boussinesq stratified fluid on an f plane incident on an infinitely long ridge indicates that the controlling dynamic parameters are the Rossby (Ro) and Froude (Fr) numbers (e.g., Pierrehumbert and Wyman, 1985). The former parameter (defined in (2.1)) measures the relative importance of the nonlinear terms to rotation while the latter (defined below in (2.59) indicates the relative importance of this nonlinearity to the stratification and hence, the strength of upstream blocking of the incident flow by the mountain barrier. As confirmed in the numerical experiments of Pierrehumbert and Wyman (1985), it is the ratio of the Rossby number to the Froude number which measures the horizontal extent upstream of the ridge over which the blocking influence extends 54 and which controls the subsequent turning of the incident flow along the mountain barrier by Coriolis effects. In this section, the scale analysis and numerical results of Pierrehumbert and Wyman (1985) are applied to the generation of coastally trapped disturbances by some forcing flow incident on the coastal mountains. The objective here is to determine whether the topographic and atmospheric characteristics of Southern Africa, Australia and North America vary to the extent of making the likely dynamic regimes different in each region. Thus, the scales adopted here are those typical of the formation stages of coastally trapped disturbances when the blocking of the low-level across-shore flow is important. Fr = U / N h m . (2.59) In (2.59), h m is the mountain height, U is a typical across-mountain velocity scale of the incident flow and N is the Brunt-Vaisala frequency. Note that for the two layer atmosphere observed during coastally trapped events, this frequency is calculated in terms of g' and the inversion layer thickness h- as N « (g'/hj)^/2. Also, note that a Froude number based on the mountain height rather than the inversion height is used because experiments (Baines, 1979) have shown that the upstream influence on the flow is sensitive only to the former parameter. Table 2.2 presents typical values of Ro, Fr and their ratio Ro/Fr from the observations of the disturbance events in North America, Australia and South Africa. To allow comparisons to be made, a nominal incident velocity of 5 m/s is chosen, which is of the order of that indicated by the data (Gill, 1977; Anh and Gill, 1981, Bannon, 1981; CLW, 1984; Dorman, 1985, 1987; Holland and Leslie, 1986; Mass and T a b l e 2 .2 Rossby (Ro) and Froude ( F r ) numbers and t h e i r r a t i o (Ro/Fr) f o r t h e c o a s t a l l y t r a p p e d d i s t u r b a n c e s o f S o u t h e r n A f r i c a , A u s t r a l i a and N o r t h A m e r i c a . S o u t h e r n A f r i c a A u s t r a l i a N o r t h A m e r i c a Ro 0 . 1 0 . 8 1 . 1 F r 0 . 3 3 0 . 2 6 0 . 2 1 Ro/Fr 0 . 3 3 . 1 5 . 3 56 Albright, 1987). Similarly, the mountain height is taken as 1000 m in each case and the halfwidth of this 1000 m contour used for the scale L. Although the coastal mountains in each area are of slightly different height, this value is an appropriate one to allow effective trapping of the inversion whose height typically varies between 400 and 1000 m (as given in the references cited immediately above). Thus, the important parameters in the comparative study of the generation of coastally trapped disturbances are the mountain halfwidth L, the reduced gravity g' (which measures the strength of the inversion), the thickness of the inversion layer hi and the Coriolis parameter f. The latter parameter shows significant variation between the three regions since in South Africa, the coastal low forms in the region 25 - 30 S, in North America, the formation region of the coastal ridges is 32 - 40 N, and in Australia, the disturbances are generated between 36 - 39 S. As discussed by Pierrehumbert and Wyman (1985), the importance of the Coriolis force is that it inhibits the blocking influence of the mountains by forcing a geostrophic adjustment of the incident flow sufficiently far upstream, typically within a Rossby radius. Satellite imagery of the initial stages of the disturbance formation in both Australia (Holland and Leslie, 1986), North America (Dorman, 1985, 1987; Mass and Albright, 1987) and South Africa (Lutjeharms, 1988 pers. commun.) confirm this theoretical result. Were this limitation of the upstream influence not so, then extensive mountain ranges like the Rockies and the Andes would have layers of blocked, stagnant fluid extending for hundreds of kilometres upstream. However, within a Rossby radius or so of the mountains the incident velocity is blocked initially to some extent so that it becomes subgeostrophic. The strength of this blocking depends on the Froude number. For Fr < 1, as in all cases in Table 2.2, 57 the initial blocking is strong but whether this upstream influence persists depends on Ro/Fr (Pierrehumbert and Wyman, 1985). Resulting from this blocking, the unbalanced along-mountain pressure gradient drives the velocity in that direction positive which in turn creates a Coriolis force in the cross-mountain direction that tends, after an inertial period, to accelerate the incident velocity back towards geostrophy. Thus, the initial minimum of this velocity reached in the first inertial period due to blocking does not last since Coriolis effects cause this velocity to return to a nearer geostrophic value. Unlike the non-rotating case, no permanent upstream modification of the incident flow occurs (Pierrehumbert and Wyman, 1985) and the accelerated along-mountain jet is trapped to within a Rossby radius of the mountain. Based on the Pierrehumbert and Wyman results,, the dynamic influence of the coastal mountains of South Africa, western North America and Australia can be interpreted as follows. The South African coastal low is potentially the simplest case since its low Rossby number (see Table 2.2) implies that nonlinearities in the early stages of coastal low formation and propagation are not important. As a result, the overall interaction of the large scale incident flow on the coastal escarpment may be considered to be within the quasigeostrophic regime; the other requirements (Gill, 1982; Pedlosky, 1987) of small Burger number and time scale of order 1/f or greater also hold. This result gives support for the quasigeostrophic models of the generation of the coastal low by Anh and Gill (1981) and Bannon (1981). In addition, the small Ro/Fr ratio for South Africa implies that the effect of blocking is not strong which in turn implies that the simple generation mechanism of coastal 58 lows via the blocking of the incident synoptic flow, suggested by Anh and Gill (1981) and Bannon (1981), is too crude to explain the details of coastal low formation. Although the large mountain halfwidth L (600 km) of the South African escarpment might imply that beta effects (i.e., those associated with the planetary vorticity gradient) are important, this has been shown not to be the case. Bannon (1981) illustrated that no Rossby waves were generated by the incident flow while Anh and Gill (1981) showed that so long as this flow was slower than about 12 m/s beta effects could be ignored. Hence, in the initial stages at least, the South African coastal low can be treated as a linear, weakly blocked system on the f plane. Contrasting this situation is the North American case which is characterised by a Rossby number of order one and a large Ro/Fr. The order one Ro implies that nonlinearities will be important while the large Ro/Fr ratio indicates that blocking of the incident flow will be strong. Also, no steady state of the incident and resulting . Coriolis turned flow along the mountain can be expected, unlike for South Africa. The relatively narrow coastal mountain halfwidth (50 km) means that beta effects in the formation of the disturbance are negligible. These results are all consistent with the energetic, nonlinear disturbances observed here and the viewpoint that these phenomena may propagate as either a gravity current or a nonlinear Kelvin wave or a combination of the two but not as a linear Kelvin wave. In the case of Australia, the dynamic parameters appear to be similar to the North American case. (Table 2.2). The Rossby number is slightly less than unity and the Ro/Fr ratio, while much larger than the coastal low case, is of the order or less than the North American value. Thus, like the North American disturbances, blocking of the incident flow is expected to be strong. However, since Ro < 1, it is possible that 59 a steady state of the incident and turned flow may eventually be reached (Pierrehumbert and Wyman, 1985). Again the halfwidth of the coastal mountains (75 km) is too small for beta effects to be important. On the other hand, nonlinearities may be significant since Ro is not much less than one. Hence, like the North American case, the generation of the Australian coastal ridging should be considered a nonlinear system with strong blocking influences. Such a view is consistent with the Holland and Leslie (1986) hypothesis that the ridges form initially as a gravity current which then evolves into a Kelvin wave that propagates through and ahead of the gravity current front. An important feature of the formation of the along-mountain winds by the Coriolis turning of the incident synoptic flow is that neither complete blocking nor geostrophic balance of these winds with the across-mountain pressure gradients are necessary. All that is needed is that the incident velocity be decelerated to a subgeostrophic value for a sufficiently long time and over a sufficiently long distance (Pierrehumbert and Wyman, 1985). Thus, the fairly wide range of parameters observed in the three different regions (see Table 2.2) allows the same qualitative behaviour for all coastally trapped disturbances in their formation stages. Having dealt with the likely dynamic regimes to be expected during the formation stages of coastally trapped disturbances, a scale analysis of the equations of motion appropriate to their propagation is now considered. b). Propagation As before, the governing equations are the shallow water equations for- a reduced gravity atmosphere (2.2a-d) and forcing in the form of an external pressure gradient 60 P is included. Following Pedlosky (1987), it is assumed that the offshore length scale 1 (in the x direction) is much less than the alongshore one L (in the y direction), so that variations in the velocity and pressure fields in the former direction are rapid while those alongshore are small. The other scales are U and V for the offshore and alongshore velocities, eL for the vertical displacement of the marine layer where e is small, 1/a for time and p fVl for the external pressure where p is the density of the marine layer. This pressure scale is chosen so that the Coriolis acceleration due to the fast O(V) flow will be of the same order as the offshore pressure gradient. Substitution into (2.2d), where h = h -f H produces: To1 allow for the possibility that the mass flux may be balanced in either horizontal aeLht +U{eLhu}x + V{eLhv}y = 0 (2.60) plane, the last two terms in (2.60) must be of equal order. Hence, the offshore velocity is constrained by (2.61) U = Vl /L (2.61) Applying (2.61) and the above scales to (2.2) then yields (2.62) 61 Two measures of nonlinearity, involving the advective Rossby number, arise out of the scaling. In the offshore equation this number (Rx) is given by (2.64) while for the alongshore direction the appropriate number (Ry) is given by (2.65). Note that the offshore Rossby number is much smaller than the alongshore one because of the assumption that 1<<L. In the cases of interest here, Rx is also much smaller than unity, as is the ratio al/fL. As a result, the alongshore velocity is in approximate geostrophic balance with the offshore pressure gradient whereas the offshore velocity is out of geostrophic balance. This condition, together with the 1<<L criterion, are the essence of the semigeostrophic dynamics characteristic of coastally trapped disturbances. Confirmation of these conditions for the coastally trapped disturbances of each region is evident in Table 2.3. In this Table, the offshore scale 1 has been formally set to R, the internal Rossby radius (2.8), and the alongshore velocity to the long gravity wave speed fR as is appropriate for Kelvin wave phenomena (Gill, 1982) or rotating gravity currents (Griffiths, 1986). Hence, it follows that the alongshore Rossby number Ry is identically equal to one. Note that the PNW column in this table refers to the gravity current disturbance reported by Mass and Albright (1987) which propagated from California to Vancouver Island. Rx = V l 2 = U fl L 2 £L Ry = V fl (2:65) T a b l e 2.3 S e m i g e o s t r o p h i c parameters f o r the c o a s t a l l y t r a p p e d d i s t u r b a n c e s of Southern A f r i c a , A u s t r a l i a , C a l i f o r n i a and the P a c i f i c N o r t h West (PNW). Southern A f r i c a L a t i t u d e 30 S 1/cr ( d a y s ) 6 R (km) 137 L (km) 3000 R/L 0.05 crR/fL 0.001 ( g , H ) l / 2 ( m / s ) 1 0 Rx 0.003 crL/fR 0.51 A u s t r a l i a C a l i f o r n i a PNW 37 S 35 N 45 N 2 3 3 327 210 168 1150 1600 864 0.28 0.13 0.19 0.018 0.007 0.007 19 19 16 0.053 0.019 0.036 0.24 0.36 0.19 63 Similar to the situation described for the incident synoptic flow, it is clear that there are substantial differences between the South African coastal low on the one hand and the U.S. and Australian disturbances on the other. Thus, the ratio R/L and offshore Rossby number Rx are smallest for the South African coastal low, which is as expected since this is the coastally trapped disturbance that best exhibits Kelvin wave characteristics. On the other hand, the relatively large value of these parameters for the U.S. and Australian disturbances is consistent with their nonlinear nature and more complex dynamics. However, in all four cases the semigeostrophic criteria of both R/L and Rx being small compared to unity are satisfied so that the theory of sections 2.2-2.6 should be a good approximation. With this in mind, it is considered that this theory is fundamental to all coastally trapped disturbances and that it is the local boundary conditions and forcing of each area that lead to the observed regional differences. These variations in local conditions are now discussed. 2.7 Variations in local boundary conditions, synoptic forcing and propagation characteristics of coastally trapped disturbances Table 2.4 (based on information given in standard Atlases) shows the topographic differences between South Africa, southeastern Australia and western North America that are of relevance to coastally trapped disturbances. The importance of the height, width and shape of the coastal mountains as well as the values of the Rossby and Froude numbers arising out of these topographic parameters has already been emphasized and so will only be discussed briefly. It was seen that the large width 64 T a b l e 2 .4 T o p o g r a p h i c d i m e n s i o n s and Rossby and Froude numbers o f t h e i n c i d e n t f l o w f o r t h e c o a s t a l l y t r a p p e d d i s t u r b a n c e s o f S o u t h e r n A f r i c a , A u s t r a l i a and N o r t h A m e r i c a . S o u t h e r n A f r i c a A u s t r a l i a N o r t h A m e r i c a Width of c o a s t a l p l a i n (km) 50-70 50-70 50 H a l f width of 1000m contour (km) 600 50-75 50 Length of l ee s l o p e (km) 900 50 50 Continent width 1800 4000 5000 (km) Ro 0.11 0.80 1.11 Fr 0 .33 0 . 26 0 . 21 65 and interior plateau of the South African landmass are favourable for linear flows whereas the narrow coastal mountains of Australia and western North America result in nonlinearities being important. Also favourable for the efficient trapping and linear propagation of the South African coastal low is the relatively smooth and consistent barrier presented by the escarpment there; an exception of course is the large bend in the southwestern Cape mountains (near 33 S, 21 E, see Fig. 2.1) about which more will be said later. On the other hand, the coastal mountains of California do have significant variations; most importantly at San Francisco Bay and Cape Mendocino, to a lesser extent at Point Arena and Point Conception (see Fig. 2.3). Further north, only Cape Blanco in Oregon appears to be significant since the gravity current case study examined by Dorman (1987) ceased propagation there, for example. Other coastally trapped disturbances of the PNW, which are often able to reach northern Vancouver Island (Mass and Albright, 1987), cease propagation as a result of dissipative influences rather than via topographic blocking. However, since these disturbances are generated usually in the Southern California Bight, the variations in the coastal mountains there are of importance as well. 1 Similarly, the pronounced bends in the Australian coastal mountains near the southeastern tip, and just north of Brisbane (see Fig. 2.2) present significant obstacles to the propagation of the coastal ridges. Thus, in the event studied by Holland and Leslie (1986) the disturbance ceased propagation at the Brisbane bend but was vigorous enough to flow around that presented by the mountains near Gabo Island. Other variations which may be important include the gaps in the coastal mountains near Williamtown (Fig. 2.2). 66 In all cases, variations in the orientation of the coastal mountains if they do not cause cessation of propagation exert a certain frictional loss on the energy of the disturbance. Hence, any model describing the evolution of coastally trapped disturbances may need to include some representation of this loss for improved accuracy. Incorporating dissipation (Grimshaw, 1983; Smyth, 1988) leads to the KdV equation (2.52) being modified to: A T + nAA T + m A T T T + F(A) = 0 (2.66) where the operator F(A) is the Fourier transform of the dissipative term that depends on (-ik)s, k being the wavenumber. In the case of viscous effects or a porous boundary (Appendix 1), s = 2 and (2.66) becomes the KdV-Burgers (KdVB) equation (2.67), whereas for boundary layer friction at either the surface or the inversion, s = 1/2. A T + nAA T -f m A T T T - | i A T T = 0 (2.67) where |x is the viscous coefficient and is positive. Analysis of these equations (Grimshaw, 1983) has shown that, asymptotically, the amplitude of the KdVB solitary wave decays as T"-*- (T being the slow time scale (2.53)) whereas in the boundary layer friction case, a faster decay of T"^ is predicted. For long enough times, Grimshaw (1983) states that the solutions become dominated by dissipation. Johnson (1970, 1972) and Jeffrey and Kawahara (1981) have shown that the KdVB equation has a steady, undular bore solution for sufficiently long times, which also occurs if localised, resonant forcing is included (Smyth, 1988). Thus, in the context of the North American and Australian disturbances, the effects of gaps in the 67 mountains can be considered by treating this boundary as porous so that (2.52) becomes the KdVB equation (2.67). Maritime influences include sea surface temperature (SST) variations due to upwelling and the local currents. Both the U.S. and South African disturbances propagate along coasts that are subjected to pronounced upwelling. Along the North American coast from California to Vancouver Island inclusive, inshore upwelling of cold water during the summer months when the prevailing winds are northerly produces SST anomalies of 5 - 10 °C typically. Similar anomalies are characteristic along the west coast of South Africa. Upwelling there, although more common during the austral summer, may happen throughout the year since the low level southerly and southeasterly winds may also occur in the austral winter (Shannon, 1986). Note that for upwelling to occur on the west coast of a Southern (Northern) Hemisphere continent, the surface coastal waters must be driven westwards and hence from Ekman theory (e.g., Gill, 1982 or any standard oceanographic text) the near surface winds must be southerly (northerly). The importance of cool SST is that they sharpen the land/sea temperature contrast and so help to strengthen and lower the inversion which favours efficient trapping of the disturbances. For example, over the North Pacific the inversion drops from about 2000 m in mid-ocean to about 400 - 500 m along the Californian and Oregon coasts (Beardsley et al., 1987). Cool currents in the North East Pacific (California Current) and South East Atlantic (Benguela Current) Oceans help in this regard. In addition, local temperature differences have an equivalent orographic effect (Fandry and Leslie, 1986) which can be seen most easily in terms of a vorticity argument. For example, a cool region compresses the vortex tubes of the fluid flowing over it and so acts like an elevation whereas a hot region is similar to a depression by causing local ascent and 68 hence vortex stretching. Modelling of this effect of surface temperature variations can be achieved parametrically through use of the adiabatic lapse rate. Then, one degree of surface cooling is dynamically equivalent to a topographic elevation of 153 m. On the eastern Australian coast and south and east coasts of Southern Africa warm ocean currents (East Australian and Agulhas Currents respectively) are found. In these regions therefore, the greater surface fluxes are expected to lift and weaken the inversion. As the data presented in the following Chapters show, the inversion is indeed higher here than on the west coasts of Southern Africa and North America. Hence, propagation speeds are generally higher on these warm water coasts. Another important difference between the three regions concerns the type of synoptic forcing responsible for the generation of the disturbances. For example (CLW, 1984), the South African coastal low is strongly tied to the weekly synoptic weather cycle. In the lower atmosphere, is common in all seasons and is relatively periodic. On the other hand, the North American and Australian coastal ridges seem to be summer phenomena (Holland and Leslie, 1986; Mass and Albright, 1987; Dorman, 1987) because this is when the subsidence inversion is best established. Also, these ridges appear to be aperiodic occurring 3 to 6 times a month. It is probable that the more seasonal and less regular nature of these disturbances arises partly from the nature of the synoptic forcing (only the South African coastal low is continuously forced throughout its propagation) and partly from the reduced maritime influence compared to South Africa. This variation in maritime influence may be seen from computations of the continentality index (defined in Barry and Chorley (1982) who state that the index ranges from -12 at extreme oceanic stations to 100 at extreme continental ones) from the climatic data in the World Survey of 69 Climatology Series (Landsberg, 1972). These computations (Table 2.5) show significant variations between the three regions, particularly in the interior. For example, in South Africa, this index ranges from negative values (-4.4 at Alexander Bay on the west coast) at many coastal stations to about 16-18 at typical interior stations (Beaufort West, Oudtshoorn). In Australia, typical coastal stations such as Adelaide and Sydney have indices of 15 and 10 respectively, while inland, Canberra and Alice Springs have values of 23 and 50. For western North America, San Diego and Los Angeles on the coast record 6 and 9 respectively, whereas Phoenix (Arizona) and Fresno (California) in the interior give indices of 49 and 35. Comparison of the values in Table 2.5 show that there is a greater maritime influence in Southern Africa than in southeastern Australia or western North America. In summary, the analysis has shown that while the fundamental dynamics of the various coastally trapped disturbances are identical (hydrostatic and semigeostrophic), there are obvious differences in the forcing and boundary conditions. As will be seen in the following chapters, these differences are substantial enough to lead to a variety of manifestations of the disturbances. 70 T a b l e 2.5 C o n t i n e n t a l i t y I n d i c e s S t a t i o n L a t i t u d e A l t i t u d e A n n u a l Temperature Index (m) Range ("C) •Alexander Bay 28 34 S - 4.5 -4 .4 Cape Town 33 54 S 9.0 9 . 4 P o r t E l i z a b e t h 33 59 s - 6.5. -0 . 6 E a s t London 33 02 s - 6.5 -0 , .1 Durban 29 50 s - 8.0 6. . 9 Umtata 31 35 s 696 10 . 0 12, . 1 Oudtshoorn 33 35 s 335 12 . 5 18 . 0 B e a u f o r t West 32 21 s 857 11. 5 16 . 1 A d e l a i d e 34 56 s - 12.0 15. o . Sydney 33 51 c - 10 . 0 10 . , 1 B r i s b a n e 27 28 c - 10 . 0 18 . 6 A l i c e S p r i n g s 23 23 s 579 16 . 5 50 . . 0 Cook 30 37 c 123 12. 6 21 . 7 C a n b e r r a 35 17 s 559 14. 7 9 San Diego 32 44 N - 8 . 4 6. 0 Los A n g e l e s 34 03 N - 9.6 8. 7 P h o e n i x 33 26 N 340 22. 5 49 . 0 F r e s n o 36 46 N 100 19 .5 35 . 0 B l u e Canyon 39 17 N 161 18.0 27 . 9 71 Chapter 3. Coastal Ridges of California and the Pacific North West 3.1. Introduction In this chapter, coastally trapped ridges that propagate in the marine layers of California and the Pacific North West (PNW) will be discussed. As mentioned in Chapter 1, both solitary Kelvin wave and gravity current models have been invoked to explain the propagation of one of these events, namely that of 3-7 May, 1982 in central California, and a vigorous dispute in the literature has arisen (Dorman, 1985, 1987, 1988; Mass and Albright, 1987, 1988). Much of this chapter will therefore be devoted towards resolving this dispute. To accomplish this, the published data (Dorman, 1985, 1987; Mass and Albright, 1987) will be re-examined in terms of the theory developed in Chapter 2 and pre-existing data (some obtained from a San Diego State University internal report, Dorman (1984), and the rest direct from Dr. Dorman), which was not used by either of these authors in their papers and which sheds further light on this dispute, will be analysed. Two other ridging events along the Californian coast will be compared with the May, 1982 event in this Chapter and considered in detail in Appendix 2. 3.2. Synoptic Description of the Californian event of 3-7 M a y , 1982 Figure 3.1 presents a set of visible images of the U.S. West Coast obtained during the event by the GOES-West satellite. The sequence begins (23 UTC 3May) with clear skies along most of the Californian coast except to the south of Point Conception (see Figs. 3.2a-b for locations of place names) in the Southern Californian 72 15 UTC 5 May 22 UTC 5 May 15 UTC 6 May 1 UTC 7 May 15 UTC 7 May Figure 3.1. GOES-WEST visible satellite imagery of the southwest U.S. and adjacent Pacific Ocean for 23 UTC May 3 to 15 UTC May 7, 1982. Adapted from Mass and Albright (1987). 73 125* 120° 50-45" 40' 35' Sounder k Eslr ronPt 69 cociMfpMI Totooshb. *| Destruction Is. * Hoquiam Astoria 46010 PACIFIC OCEAN " . - p e n I J ,1 lllSlisi^ B^III » Bend 1^0dTp ; OREGON 46022 • 46030 Cape Mendoci V ^ r V 46014- ]x> icl T ' Pt. A rena \ \ i N l 5 * 4 6 0 , 3 - \ r2CALr : fOR : f } Son Francesco ,- v s £ 4 6 0 , 2 •V?v"\ UonltrtJ LEGEND fZE3> 300m V&Zii > 1500 m LOCATIONS § Voncovrer Islond Stroll e( Juan de Fuca Georgia Strait <Q Seottle.Wo. (£> Amphitrile Pi , B C ® Vondenbero, AFB 46028 Pt. Conception 46023 125* 120' Figure 3.2a. Topographic and place location map of the west coast of the U.S. and southern British Columbia, Canada. Adapted from Mass and Albright (1987). t2S*w 120* us* Figure 3.2b. Topographic and meteorological southern Oregon. Note that only the last two are given unlike in Fig. 3.2a in which all five (1985). 74 Surface and upper air t lauonl. Symbol Type Sar&ot Station N O W 22 22 Au lonu lnd Buoy S C 1 O b K r a a o o M t t C a f c r i S o P C U n i t e d Obaenuaon Bk B a r o p i p o N M O 14 14 Automated Buojr * Point Arena pA Aufoouued Land Station Gaafab — BajUKlipfe. Al I wnut.nl St> B a t c h Wind Recorder B B Cams Guard Laad Sut ioa N D B O 1) 1) Automated Buoy N D B O 12 •2 Automated Buoy W»un Poiot TC Automated Land Station Point Pilms P N Coast Guard Land S u b o n Point Sur S U C o a a Guard Land Station Paint P f c d m P B Coast Guard Land Station N D B O II 11 Automated Buoy Vandcnberi V A N Ai r Force Base Upper air itatiom MaUbrd M F R VS. National Weather Service Oakland O A K VS. rUi ional Weather Service V u d t n b e r f A F B V A N VS. A i r Weather Service S D VS. National Weather Guadalupe bland - M r i i c a n Weather Service station location map of California and digits of each buoy number (as in the text) digits are given. Adapted from Dorman 75 Bight where a localised area of stratus cloud is beginning to form in response to the forcing. Over the next few days the stratus was observed to spread northward up the coast as a narrow tongue until by 15 UTC 5 May it had reached Cape Mendocino (see Fig. 3.2a), the point of furthest advance. Small cyclonic eddies (30 km and 120 km in diameter respectively) were observed to form at the Cape and just to the south at Point Arena (see Fig. 3.2a). These features were observed again on 6 May (15 UTC) but over the following day (01 UTC and 15 UTC 7 May) a retreat of the stratus occurred which was accelerated as a weak front moved southeastwards into the region. The evolution of the synoptic and mesoscale pressure charts at 850 mb and at sea level that accompanied the event are shown in Figs. 3.3 and 3.4a-b respectively. Initially (00 UTC 3 May - 00 UTC 4 May), strong northerlies dominated the northern Californian coast while weaker northwesterlies and westerlies prevailed along the southern coast. Between 00 UTC 4 May and 12 UTC 4 May a synoptic scale low passed at 700 mb across southern California. It is the interaction of this synoptic feature with the coastal mountains that forces the initial raising of the marine layer in the Southern California Bight from which the disturbance developed (Dorman, 1985, 1987). Associated with the low are onshore winds in the Bight (Fig. 3.3), which lead to the disturbance forming as a coastal ridge rather than as a trough (see Section 2.4). As a result of this forcing, a mesoscale surface ridge developed in the marine layer which, as it propagated northwards from just south of Point Conception to Cape Mendocino, caused a shift in the nearshore winds to a southerly flow that was restricted to a distinct, narrow zone seawards of the coastal mountains which was shown to be approximately a Rossby radius in width (Dorman, 1985; Mass and Albright, 1987). Soundings (Figs. 3.5a-b) taken in the Southern Californian Bight 76 1_1 JL I \ I I I 1 1 00 GMT 7 MAY 1982 00 GMT 8 MAY 1982 Figure 3.3. National Meteorological Center 850 mb synoptic maps for the period 00 UTC May 3 to 00 UTC May 8, 1982. Geopotential heights (solid) are in decametres and isotherms (dashed) are in degrees Celsius. Adapted from Mass and Albright (1987). 0 0 G M T 7 M A Y 18 G M T 7 M A Y Figure 3.4a. Mesoscale sea level pressure analyses and surface winds for the period 00 UTC May 4 to 18 UTC May 7, 1982. Solid lines are sea level isobars (lOxx mb) and winds are in knots. Heavy dashed lines are the northern boundaries of the coastal stratus as defined by satellite visible imagery. Adapted from Mass and Albright (1987). 78 OO GMT 7 MAY 1982 0 0 GMT 8 MAY 1982 Figure 3.4b. National Meteorological Center surface synoptic maps for the period 00 UTC May 3 to 00 UTC May 8, 1982. Solid lines are sea level isobars (lOxx mb). Adapted from Mass and Albright (1957). 2/&(o) jyson - 2 0 - 1 0 0 10 TEUP (DEC C ) V S O I ) - 3 0 - 2 0 - 1 0 0 T t u P (DEC C ) 5 /5 (0 ) - 4 0 - 3 0 - 2 0 - 1 0 0 10 20 TCVP (DEC C ) 6 / 5 ( 1 2 ) 7 3 / 5 ( 0 ) 79 - 3 0 - 2 0 - 1 0 0 T E U P (DEC C ) '/KO) - 2 0 - 1 0 0 10 TEUP (DEC C) 4 / S ( 1 2 ) - 2 0 - 1 0 0 10 "20 TEUP (SEC C) 5 / 5 ( 1 2 ) - 2 0 - 1 0 0 TEUP (DEG C) a 3500 8 " 3000 • 3 2500 s » . 7000 § 1 5 0 0 « 1000 X 500 0 -40 - 3 0 - 2 0 - 1 0 0 TEUP (OEC C) 10 » / 5 ( 0 , *\ i ««-> - 3 0 - 2 0 - 1 0 0 TEUP (DEC C ) 7 / K 0 ) 4000 f 3500 3000 I • 2 2500 * , 2000 1 1900 U 1000 • I 500 • 0 - 3 0 - 2 0 - 1 0 0 10 T E U P (OEC C) 20 - 2 0 - 1 0 0 10 TEUP (DEC C) V5(12> - 3 0 - 2 0 - 1 0 0 10 TEUP (DEC C ) Figure 3.5a. Temperature (right hand curve) and dewpoint temperature (left hand curve) soundings through the lower atmosphere for San Diego for the period 00 UTC May 2 to 12 UTC May 7, 1982. The base of the inversion is indicated by an arrow on each sounding. 80 3 / 3 0 0 3/3(22) -10 0 10 TEUP (0£G C) */S(22) 4/5(12) -10 0 TEMP (DEC C) -20 -10 0 TEMP (DEC TEMP (DEC C) TEMP (DEG C) TEMP (DEG C) Figure 3.5b. Temperature (right hand curve) and dewpoint temperature (left hand curve) soundings through the lower atmosphere for Point Mugu for the period 11 UTC May 3 to 11 UTC May 7, 1982. The base of the inversion is indicated by an arrow on each sounding. 81 (the generation region of the disturbance) at San Diego and Point Mugu (see Fig. 3.2a for locations) show the marine layer depth to be about 1-2 km generally. Further north at Vandenburg AFB near Point Conception, Fig. 3.5c shows the marine layer depth associated with the propagation north of the disturbance to be 500 - 800 m, i.e. below the summit level of the higher coastal mountains (see Figs. 3.2a-b). Accompanying the propagation of the coastal ridge were substantial drops in temperature as subsiding warm continental air was replaced by cool maritime flow and overhead, the northward progression of the stratus deck. Sea level pressure then rose over most of the region during the next twelve hours (to 12 UTC 5 May) as cooler air and associated higher surface pressure moved southwards into central California behind a cold front. By this time the southerly winds and stratus had progressed as far as Cape Mendocino where a cyclonic eddy formed. A larger eddy also formed at Point Arena some 100 km to the south. Over the next 24 hours, a diurnal cycle of dissipation of the stratus during the day followed by thickening at night occurred but no further movement of the cloud northwards was observed. It appears that the sharp convex bend formed by the coastal mountains at Cape Mendocino prevents further propagation of the stratus up the coast, a process that can be simply explained (Reason and Steyn, 1988) if the solitary Kelvin wave model is valid. After 12 UTC 6 May, the Pacific Anticyclone retreated away from the Pacific North West resulting in a fall in pressure over the entire southwestern U.S. and in turn the recurrence of a strong alongshore pressure gradient and northerly flow over northern California. As a result the stratus deck retreated to the south and dissipated over the following two days. 2/5(0) -20 - 1 0 0 10 TEUP (DEG C) 3/5(12) 2/5(12) 8 2 -20 -10 0 TEUP (DEC C) 4/5(0) - 2 0 - 1 0 0 10 TEUP (DEC C) 3/5(0) - 2 0 - 1 0 0 10 TEMP (DEG C) «/3(12) - 2 0 - 1 0 0 10 TEUP (OEC C) - 1 0 0 10 TEUP (OEC C) 5/5(0) 3/5(12) -5 0 S 10 TEMP (OEC C) «/5(i2) 6/5(0) 4000 -(D a 3500 • 2 8 3000 -1 1 J5 2500 • cn »•*» Z, 2000 -- J - i 1500 • / » -I § 1000 -/ u X 500 -/ 0*--30 -20 -10 0 TEUP (DEC C) - 2 0 - 1 0 0 10 TEUP (DEG C) - 2 0 - 1 0 0 10 TEMP (DEG C) 7/5(0) V3(12) 4000 4000 'v 5 3500 8 a 3500 1 3 3000 l 3000 • \ A 3 2500 IN i 2000 2. 1500 • J 7 2 2500 S Z. 2000 2- isoo \ Vs 5 IOOO 1 1000 \. UJ X 500 • i 500 \ X c -30 -20 -10 0 10 20 -30 - 20 -10 0 10 20 TEUP (DEG C) TEMP (OEC C) Figure 3.5c. Temperature (right hand curve) and dewpoint temperature (left hand curve) soundings through the lower atmosphere for Vandenburg for the period 00 U T C May 2 to 12 U T C May 7, 1982. The base of the inversion is indicated by an arrow on each sounding. 83 3.3 Solitary Kelvin Wave or Gravity Current ? Some detailed measurements taken from Dorman (1985) and Mass and Albright (1987) are now discussed in an attempt to understand the dynamics of the phenomenon in terms of the solitary Kelvin wave or gravity current models. Upper air soundings (Figs. 3.5c, 3.6) from Vandenburg AFB show the height of the marine layer (i.e. the inversion) increasing from a minimum at 1200 UTC 3 May to a maximum of 843 m a day later. At Oakland, Fig. 3.6 also shows the inversion height to be a minimum at 0500 UTC 3 May before reaching a maximum of 600 m about 2 days later. Note that both stations exhibit the expected nocturnal reduction in the inversion height. Fig. 3.7 shows the surface pressure and wind changes associated with the passage of the disturbance at, Gualala. Also shown in this figure are these data for two buoys (C3 and C5) located just offshore of Gualala (see Fig. 3.8b for location of C5; buoy C3 is situated between buoy C5 and Gualala). The shape of the pressure variation at Gualala (Figs. 3.7 and 3.8a) is entirely consistent with that of a solitary wave and completely unlike that recorded at this station during the gravity current event (Dorman, 1987) of 14-20 July, 1982. Fig. 3.8a, which compares surface pressure data measured at various coastal stations during the May and July, 1982 events, shows that the pressure variation at Gualala caused by the gravity current event of July, 1982 to be unsteady and long term (almost a week), which is as expected for a gravity current (see Chapter 2). It is clear from Fig. 3.8a that the contrast in surface pressure variation between the two types of event observed at Gualala is also evident at several other stations along the central Californian coast. Consistent with the hypothesis advanced in this chapter, the pressure data for the May, 1982 event are all typical of a solitary wave while those for the July, 1982 case all display the features expected for a gravity current. Figure 3.6. Height of the base of the inversion at Vandenburg and Oakland for the period 00 UTC May 2 to 12 UTC May 8, 1982. The inversion lifted and descended at Oakland 24 hours after it did at Vandenburg. Adapted from Dorman (1985). MAY 3 4 5 6 7 — I — I — i — | — r ~ i — i — j — r n — i — | — i — i — i — l — i — i — i — r N O R T H WIND + 10 o v> >* E - i 0 •io - E A S T WIND J — i — i — i — i — i i — i i i i i i i i i • • • > Figure 3.7. Surface winds and pressures for Gualala station and offshore buoys (C3, C5) on the central Californian coast during the May, 1982 event. The winds have been rotated so that "North" is actually 317* which is aligned with the coast. Station Sea Ranch is located on the coast a few kilometres south of Gualala. Adapted from Dorman (1985). 85 39 N 12-4/W o NDBO 14 PT. ARENA LIGHT GUALALA PT. Mk W 38 N i) ELK P.O. 39 N u ' u 1 . . 1 . . 1 . . 1 . . 1 . . 1 . . 1 . 1 . , I . 1 . 1 FORT ROSS BODEGA BAY B 38 N Figure 3.8a. Surface pressure traces for various stations along the central Californian coast for the May, 1982 (left hand plot in each case) and July, 1982 (right hand plot) events. On the vertical axis of each trace, the numbers (xx) refer to lOxx millibars while the horizontal axis gives the day in either May or July. 124 W 86 124 W 38 N 7 O NDBO 14 2 £< PT. ARENA LIGHT GUALALA PT. 39N o WHOI C5 FORT ROSS a 1 1 . 1 » 1 •• 1 w 1 II 1 i. 1 » 1 • ' « 1 • 1 » 1« o NDBO 13 'BODEGA BAY ^MARINE 0 LAB Figure 3.8b. Alongshore surface pressure differences (in millibars) for various stations along the central Californian coast for the May, 1982 (left hand plot in each case) and July, 1982 (right hand plot) events. In each plot, the horizontal axis gives the day in May or July. 1 = buoys 22 (not shown, but located 50 km north of buoy 14, see Fig 3.2a) - 14, 2 = buoys 14 - 13, 3 = Point Arena - Fort Ross, 4 = buoys C5 - 13. Note that 1 (obtained from Dorman (1985)), is plotted with the scale on the vertical axis reversed and that this plot as well as plots 2-4 are such that a positive pressure difference corresponds to a northerly wind for downgradient flow. 87 Another way of seeing whether the low level wind and pressure changes along the coast are wavelike in form, is to plot the observed surface pressure gradient differences that occurred between sets of coastal stations. Dorman (1984, 1985) has calculated these differences for various coastal stations and buoys (Fig. 3.8b). Again, there is a clear distinction between the data for the May, 1982 case (coherent and pulse like) and that for the July, 1982 gravity current event (unsteady and variable). For buoy pair 22-14, which is situated astride Cape Mendocino (see Fig. 3.2a) north of Point Arena, the large pressure difference for the May, 1982 event is associated with the cessation there of the northerly progression of the solitary wave (Dorman, 1985). Further south, the May, 1982 record from buoy pair 14-13 (and similarly for that from station pair Point Arena - Fort Ross and from buoy pair C5-13) shows strong northerly winds initially followed by a reversal to southerly between 1500 and 1800 UTC 4 May as the leading edge of the wave arrived. The reversal in pressure difference and wind direction (to northerly again) after about 1700 UTC 6 May indicates the passage of the trailing edge of the wave. Further support that the negative pressure difference which existed during May 4-6 is accounted for by the lifting of the marine layer is provided through use of equation (3.1). Note that this lifting of the marine layer is caused directly by the passage of the wave along the layer. P = P'exp(gH/287.04Tv) (3.1) In (3.1), P is the pressure at the bottom of the marine layer, P' that at the top, H is the layer thickness and Tv is the (virtual) temperature of the layer. From observations, the inversion height at the coast increased from about 100 m on the afternoon of May 4 to about 500 m later that day as the leading edge of the stratus 88 cloud passed. Thus, if a layer 400 m thick with a virtual temperature of 25 °C were replaced by one at 15 °C due to the passage of the disturbance, then the surface pressure would increase by 1.6 mb which is close to the 2 mb observed for buoy pairs 14 - 13 or C5 - 13 (Dorman, 1985). Also consistent with the solitary Kelvin wave model, are the negligible across-coast surface winds measured at Gualala and buoys C3 and C5 during 4-6 May, 1982 (Fig. 3.7). As discussed in Chapter 2, if significant across-shore flow is induced at the leading edge of the disturbance then there can be no dispersion and hence a solitary wave can hot exist. Instead, the coastal (nonlinear Kelvin) wave associated with the gravity current steepens and ultimately forms a shock wave. While this does not happen for the May, 1982 case (Figs. 3.7 and 3.9), significant offshore winds are present at the leading edge of the July, 1982 (Fig. 3.10) and May, 1985 (Figs. 3.11, 3.12) gravity current events. Another significant feature of Fig. 3.11 is that it shows the abrupt change in surface temperature (almost 20 °C in under an hour) associated with the gravity current front. Such abrupt changes were not reported during the May, 1982 event by either (Dorman, 1985, 1988) or Mass and Albright (1987, 1988) despite the fact that similar hourly resolution buoy data was available at several locations during this event. As discussed in Section 2.3, such abrupt changes in the surface or marine layer temperature are not expected for Kelvin waves but are typical of gravity currents. Further reinforcement for the solitary Kelvin wave model of the May, 1982 event is evident in the satellite images of Figs. 3.13 - 3.14. These figures show that the stratus cloud deck associated with the initial raising of the marine layer was localised within a narrow coastal area in the Southern California Bight. Soundings from San Diego and Point Mugu (Figs. 3.5a-b) have indicated that this raising of the marine layer at 89 22 SC PC 14 PA 13 BB 12 PG PN SU-PB 11 rrr/rr rrrffrrrrn-fr vUmurn- rrr rrrr i \ / F- r r ."D ITT w v < u s , r r r ^ r/ . h \ . . . v m t t n r r r rrrrrrFf-r / /JJ/JJJ \Ui| yjJUJjjJ^ / y ^ ^ ^ ^ ^ ^ / 1 1 i 4 1 1 1 I 1 1 I I 1 1 1 I I 5 6 7 MAY Legend / * A b. \ c. t <• t f. • g h. Figure 3.9. Surface winds at stations (see Fig. 3.2b for locations) along the Californian coast. Wind speeds are given in m/s and dots indicate calm. See legend for wind barbs in m/s: a) 1. b) 5, c) 2.5, d) 12.5, e) gusting to 10, f) 7.5 m/s gusting to 15 m/s, g) calm, h) a wind from the north and from the southwest. Adapted from Dorman (1985). 90 Figure 3.10. Surface pressure and winds along the northern Californian coast (just south of Cape Mendocino) recorded at 1800 UTC, July 17, 1982. The leading edge of the gravity current has reached Point Arena where significant across-shore winds are observed. The wind vectors point downstream. Adapted from Dorman (1987). 91 ASTORIA, OREGON 20 21 22 23 0 I 2 3 4 9 6 7 6 9 KJ II 12 . M A Y 16 M A Y 17 T I M C ( G M T ) Figure 3.11. Surface temperature, winds and pressure recorded at Astoria, Oregon during the May, 1985 gravity current event. Significant across-shore (NE and W) winds are observed. Adapted from Mass and Albright (1987). 92 Figure 3.12. Mesoscale sea level pressure analyses and surface winds for the May, 1985 event. The heavy dashed line marks the leading edge of the gravity current. With the exception of the first and last panels, significant across-shore winds are observed near each leading edge. Adapted from Mass and Albright (1987). Figure 3.13. G O E S - W e s t satellite visible imagery for 0045 U T C M a y 3, 1982. The localised ini t ia l displacement (low level stratus deck) of the marine layer in the Southern Cal ifornian Bight is evident. Note the t rapping by the coastal mountains. Cour tesy of D r . Dorman . Figure 3.14. GOES-West satellite visible imagery for 2315 U T C M a y 3, 1982. Some erosion of the stratus cloud due to d iurna l heating is evident on comparison of the s tratus deck with that shown in Fig. 3.13. Note that U T C is 7 hours ahead of local time. Cour tesy of Dr . Dorman . 94 the coast was pronounced. Thus, it appears that the closed low provided a concentrated forcing of the marine layer within a narrow coastal zone, i.e., in analogous terms, acted somewhat like an impulse wave-maker in a laboratory tank. The evolution of the marine layer disturbance in the Bight into a propagating solitary wave has been discussed in Chapter 2 and will be dealt with in more detail below. It is clear however, that this initial disturbance is too small to form the reservoir necessary for coastal gravity current propagation (see Chapter 2). On the other hand, the passage of the baroelinic trough at 700 mb, that forced the July, 1982 event, caused a widespread and less concentrated raising of the marine layer in the Southern California Bight (Figs. 3.15a-b) which could act as a substantial reservoir for coastal gravity current flow. Finally, it is shown that the observed speed of propagation of the May, 1982 disturbance is best matched by the nonlinear Kelvin wave phase speed. From Dorman (1985), the observed speed of propagation of the disturbance northwards was 6 m/s along the central Californian coast while the reduced gravity g' of the coastal atmosphere was 0.41 ms"2 and the mean height H of the marine layer at Gualala station (on this coast) was 400 m. Thus, the linear Kelvin wave phase speed was 13 m/s. Opposing this is a mean northerly flow of 7.5-10 m/s at Gualala which then gives a net propagation speed of 3.0-5.5 m/s as compared to the 6 m/s observed. The maximum probable error in the computed linear Kelvin wave speed may be calculated as ±1 m/s from the data of Dorman (1985). Better agreement is achieved using the solitary Kelvin wave speed: c = {g'H(l + a)} 1/ 2 (3.2) Figure 3.15a. GOES-West satellite visible imagery for 1616 UTC July 12. 19S2. Note the reservoir formation (widespread stratus cloud) in the marine layer of the Southern Californian Bight and the trapping of the northern propagating part of the stratus to a narrow coastal zone. The leading edge of this stratus deck has reached Monterey Bay just south of San Francisco. Courtesy of Dr. Dorman. 96 Figure 3.15b. GOES-West satellite visible imagery for 2115 UTC July 12. 1982. The much wider spread displacement (indicated by stratus cloud deck' of the marine layer (and hence reservoir formation) in the Southern Californian Bight than for Figs. 3.13-3.14 is evident. Courtesy of Dr. Dorman. 97 The ratio of the solitary wave amplitude to the marine layer depth a was calculated by Reason and Steyn (1988) from the data of Dorman (1985) to be 0.161 (maximum possible error ±0.010). Applying (3.2) then gives a theoretical phase speed for a solitary Kelvin wave at Gualala of 14 m/s (error range ±1.2 m/s) and hence a net propagation speed of 4.0-6.5 m/s which compares more favourably with the observed 6 m/s. On the other hand, the empirical atmospheric gravity current speed formula (2.11) of Seitter and Muench (1985) that was used by Mass and Albright (1987) to compute the theoretical propagation speed for the May, 1985 gravity current case, but interestingly enough not for the May, 1982 case, gives speeds significantly below those observed. The surface pressure difference 8P between the gravity current head and the ambient atmosphere required in (2.11) can be inferred from Dorman (1985) and Mass and Albright (1987) as being between 100 and 200 Pa. Applying (2.11) then gives theoretical gravity current speeds of 7.4 - 10.4 m/s (maximum possible error ± l m / s ) which implies that the net propagation speed is 0-2.9 m/s, a range that is considerably less than the observed 6 m/s. To summarise, observations of the surface pressure and wind variations, the initial disturbance in the Southern California Bight and the propagation speed of the disturbance are all consistent with a solitary Kelvin wave and not with a gravity current. In addition, several nonlinear features of the disturbance such as the cessation of propagation at Cape Mendocino and the decay of the amplitude of the disturbance as it propagated northwards can all be explained in terms of solitary wave dynamics (see the following discussion and Reason and Steyn, 1988). Finally, note that the observed width of the stratus deck and the zone within which the 98 surface winds and pressures were altered by the wave was well approximated by the computed Rossby radius (Dorman, 1985). 3.4 Solitary Kelvin Wave Model of the M a y , 1982 case In this section, the theory of in Chapter 2 will be applied to the Californian marine layer disturbance of 3-7 May, 1982. It has been illustrated above that the data support a solitary Kelvin wave model of this disturbance, so attention here will be focused towards showing that the theory behind this model can account for the observed time and displacement scales during the formation of the disturbance and for the observed features of propagation. Also, recall that the solitary Kelvin wave speed (3.2) agreed well with the observed speed of propagation. Until now, it has been tacitly assumed that the disturbance belongs to the KdV or shallow water theory of solitary waves. Other possible theories that allow solitary wave solutions are the Benjamin-Ono theory for a waveguide abutted by an infinitely deep fluid (Benjamin, 1967; Ono, 1975) and the finite depth theory of Joseph (1977) and Kubota et al., (1978). These alternative theories are mentioned because it is possible that, despite the initial marine layer disturbance being a shallow water phenomenon, the evolving solitary wave disturbance may fall into another category. It is shown now however, that the KdV theory is the most appropriate for the May, 1982 event. A requirement of the Benjamin-Ono theory is that the wavelength of the disturbance be much less than the total fluid depth. Since the wavelength is of the order of 1600 km (Table 2.3), this condition is never satisfied by any possible depth scale in the atmosphere. Hence, the Benjamin-Ono theory can immediately be ruled out. On the 99 other hand, the inverse condition, i.e., a wavelength much larger than either the depth scale of the density stratification or the total fluid depth is necessary for the finite depth and KdV theories respectively. Thus, on the basis of the wavelength of the May, 1982 disturbance, both theories are possible. To discriminate between these two possibilities, this stratification depth scale must be compared with the fluid depth. For an oceanic case, this stratification scale would be the thermocline thickness (Koop and Butler, 1981) whereas for the situation here, the inversion height is appropriate. From soundings at Vandenburg (Figs. 3.5c, 3.6), this height is about 800m. The finite depth theory requires this depth to be much less than the fluid depth whereas in the KdV case, the two depths should be similar. Unfortunately, it is not unambiguous what the fluid depth should be. Possible choices are the height at which the forcing occurs and at which there is a second inversion (about 3000 m, see Fig. 3.5a, 3.5c) or perhaps the density scale height of the atmosphere (8500 m). Despite this lack of clarity, neither of these choices is sufficiently deep for the finite depth theory to be appropriate since, from the experiments of Koop and Butler (1981), it is known that even for a fluid depth 36 times larger than the stratification scale this theory proved to be less accurate than the KdV theory. This latter theory was shown to be remarkably robust even outside the strict limits of its applicability. This robustness of the first order KdV theory has been experienced in other applications (Kao et al., 1985; Grimshaw, 1988, pers. commun.) and may partly arise through the higher order KdV theories (e.g., third order) also having a sech^ x solution (Gear and Grimshaw, 1983). Thus, it would seem that the KdV theory offers the best available description of the solitary wave behaviour during the May, 1982 event. 100 Following the concepts developed in Chapter 2, it is considered that the solitary Kelvin wave evolved from the initial disturbance that was observed in the Southern Californian Bight (see Figs. 3.13-3.14) on a time scale T given by (2.53). Substituting the definition of the parameter d into (2.53) yields: T = (RH/La)" 5f 1 (3.3) From Dorman (1985), R = 150 km at Gualala while a/H = 0.161 here (Reason and Steyn, 1988). Hence, (3.3) indicates that the solitary Kelvin wave could evolve from the initial disturbance in the Southern Californian Bight over a time period of about 46 hours (with maximum possible error range of ±4 hours). Such a value is consistent with the observations of Dorman (1985) (reproduced here as Fig. 3.6) which show the inversion level at Vandenburg AFB (just north of the Bight) to reach a maximum over about 36 hours after the initial raising of the marine layer. Also, the satellite images show the initial disturbance (generated around 0045 UTC May 3, Fig. 3.13) still within the Bight at 2315 UTC on 3 May (Fig. 3.14) but by 1515 UTC the next day (Fig. 3.16) the ridge had already propagated 100 km or so northwards. Thus, from these images, an evolution time of 23-38 hours is expected. Note that (3.3) is sensitive to both the ratio R/L and the nonlinear parameter a/H. Hence, solitary Kelvin waves which have larger R/L and amplitudes, will evolve more quickly. A comparison of the observed wavelength of the disturbance with that predicted by the Kubokawa and Hanawa (1984) solitary Kelvin wave solution can be made. From Fig. 3.7, the time span between the leading and trailing edges of the wave is 3.1 days and hence the wavelength or solitary wave width is given by this time multiplied by the net propagation speed of the wave. From these values, the wavelength L g of the 101 Figure 3.16. GOES-West satellite imagery for 1515 UTC May 4, 1982 showing the propagation of the stratus deck to Monterey Bay south of San Francisco (see Fig. 3.2a for location of place names) about 100 km north of the Southern Californian Bight. Courtesy of Dr. Dorman. Figure 3.17. GOES-West satellite visible imagery for 1615 UTC May 5, 1982 showing the separation of the stratus overcast from the coast at Cape Mendocino (upper arrow). Eddy formation here and at Point Arena (lower arrow) is evident. Adapted from Dorman (1985). 102 wave is found to be about 1607 km (maximum possible error ±100 km). Theoretically, L g is given by (2.55), which after substitution of a = 0.161 gives a non-dimensional wavelength of 1.22, and hence, a dimensional wavelength of 1800 km. This value is seen to agree reasonably with the observed 1607 km. The magnitude of the marine layer displacement by the synoptic forcing (closed low at 700 mb) is now considered. An estimate of this magnitude can be obtained by treating the marine layer as an Ekman layer with geostrophic drag coefficient C^g and using (3.4) given in Gill (1982). W g « (C d g/p 2f 3)|dp/dr|(2d 2p/dr 2 + 1/r dp/dr) (3-4) In (3.4), W g is the vertical Ekman suction velocity, p is the density of the air just outside the marine layer, f is the Coriolis parameter and dp/dr is the pressure gradient also evaluated just outside the layer. Note that (3.4) is, strictly speaking, applicable to steady geostrophic flow on an f plane with constant drag coefficient and surface wind direction (and hence stability). Only the first of these restrictions is of potential significance here because the other two are satisfied by the marine layer being stable over the entire region. Although the flow due to the passage of the closed low is likely to be geostrophic in the alongshore direction, the across-shore component is probably ageostrophic. However, as substitution of the appropriate values for the parameters will show, (3.4) gives a good estimate of the forced increase in the marine layer height. Since the pressure gradient just outside the marine layer is required, that observed at 850 mb rather than at the surface is used. From Fig. 3.3, this pressure gradient is of the order of 2.0 mb/100 km. Following Gill (1982), a geostrophic drag coefficient of 103 0.0001 is assumed for the stable marine layer. Substitution of these values and a Coriolis parameter for 32 N (Southern Californian Bight) into (3.4) then gives an Ekman suction velocity of 0.0175 m/s and hence, over a 12 hour period, the marine layer can be estimated as being raised by 760 m (maximum possible error ±60 m). This estimate compares favourably with the sounding data (Fig. 3.5a-c) for San Diego, Point Mugu and Vandenberg AFB. For example, at Vandenburg near the start of the propagating event proper, Fig. 3.5c shows the marine layer to be raised from near the surface to about 500 m within 12 hours and to 850 m over the following 12 hours. Further south at Point Mugu, which is about halfway along the coastline of the Southern Californian Bight, the marine layer is seen (Fig. 3.5b) to rise over 500 m from 1080m at 2200 UTC on May 3rd to 1587 m at 1200 UTC on May 4th. At San Diego, which is at the southern end of the Bight and almost directly beneath the passage of the closed low, the marine layer rises about 1500 m between 00 UTC and 1200 UTC on May 4th (Fig. 3.5a). It is clear therefore that the pressure gradient associated with the passage of the closed low is strong enough to force the raising of the marine layer to the height observed. A particularly prominent aspect of the May, 1982 event was the observed inability (see Fig. 3.17 above) of the disturbance to propagate around the sharp convex bend in the coastal mountains at Cape Mendocino. Instead, the flow is observed to separate from the coast and form a small cyclonic eddy, of about 30 km radius, just to the south of the Cape as well as a larger cyclonic eddy, of about 120 km radius, at Point Arena, some 100 km to the south. Dorman (1985) considered the disturbance to be a linear Kelvin wave and so on the basis of the diffraction theory for such a wave (Miles, 1972; LeBlond and Mysak, 1978) expected the disturbance to ordinarily be able to propagate around the Cape. To explain the observations on this basis, Dorman was forced to speculate that this cessation of propagation was due to the 104 opposing mean northerly winds north of the Cape being of speed similar to the wave phase speed of the wave. As shown below, this speculation does not appear likely. Firstly, the observed winds measured at buoy 22 just north of the Cape (see Fig. 3.9) decrease from 12.5 m/s twelve hours or so prior to the arrival of the leading edge of the wave, to 0-5 m/s as the wave passes overhead. Over the following day, the winds increased to 5-10 m/s. On the other hand, the theoretical phase speed of the solitary wave was computed at Gualala (see section 3.3 above) to be 14 m/s. Although this station is about 150 km to the south of Cape Mendocino and the disturbance speed is observed to decrease slightly as it moves north, it is improbable that the solitary wave speed at the Cape will have been reduced by the 50 % or so necessary for Dorman's blocking hypothesis to work. Instead, as is shown below, it is the size of the bend in the coastal mountains at the Cape which prevents further propagation for a solitary Kelvin wave there. The hypothesis of increased winds north of the Cape and an unrealistic decrease in the wave speed is therefore no longer needed. Miles (1977) showed that a solitary wave is unable to propagate around a convex bend in the coastline of angle greater than the critical angle $ given by : $ = (3a)1/2 where a is the ratio of the wave amplitude to the fluid depth which was calculated from Dorman's data as 0.161 by Reason and Steyn (1988). If the angle of the coastline is indeed greater than then the solitary wave must separate from the coast or otherwise lose its identity (Miles, 1977). (3.5) 105 Substituting a = 0.161 into (3.5) gives a critical angle for the bend in the coastal mountains of 39.8°. Since the bend of the 1000m contour at Cape Mendocino is of the order of 50°, it is clear that the solitary Kelvin wave should in fact not be able to propagate around the Cape but instead separate (see Reason and Steyn, 1988). Interaction of this separated flow with the mean northerly winds could then lead to the observed eddy formation through some form of baroclinic or mixed barotropic-baroclinic instability. Note that the instability studies of coastal currents and frontal waves by Stern (1980), Stern et al.,(1982), Killworth and Stern (1982) and Paldor (1983, 1988) are not valid here as these theories assume zero potential vorticity (see Chapter 2 for a discussion of this point). Another interesting aspect of the observations is that the disturbance was seen to decrease in amplitude as it propagated northwards (Dorman, 1985). This behaviour was attributed by Dorman to either energy radiation or to turbulent drag. The latter mechanism seems more likely in light of the results of some rotating tank experiments performed by Maxworthy (1983). In these experiments, it is observed that the amplitude of a propagating internal solitary Kelvin wave decays exponentially in time as a result of the drag exerted by inertial waves generated in the homogenous fluid above and below the solitary wave. Grimshaw (1985) attempted to explicitly analyse the observed decay of Maxworthy theoretically. However, not much success was achieved due to the experiments being in the strongly nonlinear regime for which the theory breaks down. An alternative hypothesis is that the observed amplitude decay is due to dissipative losses either through viscous effects within the marine layer itself or leaks through gaps in the coastal mountains (e.g., at San Francisco Bay) or through boundary layer friction at either the surface or the inversion layer. As.discussed in Chapter 2, viscous effects and gaps in the mountains lead to a T (T being the slow time scale (2.53)) amplitude decay whereas boundary layer friction causes a T"^ decay. Unfortunately, with the data available (Dorman, 1985; 1987; Mass and Albright, 1987 and Sections 3.2-3.4), it is not possible to determine which type of dissipative effect is most likely. However, on the basis of what is available, it would seem that the KdVB dissipation with its slower T"^ amplitude decay offers the most realistic description. The justification for this statement is simply that the observed solitary wave amplitude decay between 3 inshore buoys (data derived from that in Dorman, 1984) can be shown to be fairly small (0.2m/km) and roughly constant. The buoys in question are all about 20 km offshore and are 60 - 80 km apart so that as a whole they are located over the final 150 km of the propagation path of the disturbance. Note that the surface pressure records at these buoys, from which the solitary wave amplitude was calculated, are similar to those analysed above in Section 3.3 (Fig. 3.8a). In summary, it has been shown that a solitary Kelvin wave model of the May, 1982 disturbance can account for the salient features of the observed propagation. These features are the trapping of the disturbance within a Rossby radius of the coastal mountains, the displacement profiles of the surface pressure fields and inversion height, the inability of the disturbance to propagate around Cape Mendocino and associated eddy formation, the decay of amplitude as the disturbance propagated northwards and the good agreement obtained between the theoretical phase speed of the solitary Kelvin wave model and the observed propagation speed. 107 3.5 Conclusion In this Chapter, coastally trapped disturbances, which take the form of mesoscale ridges of high pressure and which propagate northwards in the marine layers of California and the PNW (Oregon, Washington and Vancouver Island), have been examined. One event (May, 1982) was analysed in detail; and two others are analysed in Appendix 2. The events considered were chosen by the original workers (Dorman, 1985, 1987; Mass and Albright, 1987) as being representative of coastally trapped disturbances in these areas, which on average occur about 4 to 5 times a month during the summer season only (Mass et al., 1986; Dorman, 1987). Note that in the context used by these authors, summer is defined as that half of the year when the North Pacific Anticyclone is the dominant synoptic feature in the region and the prevailing winds along the Californian and PNW coasts are northerly. During the lifespan of a given coastally trapped event, the switch of these northerlies to southerlies in the marine layer is one of the prominent weather changes caused by the northwards propagation of the disturbance (Dorman, 1985, 1987; Mass and Albright, 1987). Other weather changes are the raising of the marine layer and hence the overhead formation of a stratus cloud deck as the disturbances propagates and the perturbation of the pressure fields in the lower atmosphere to form a mesoscale ridge. Consistent with the theory of Section 2.4, the forcing in each case caused onshore flow in the generation region and hence coastal ridges rather than troughs were formed. In the cases examined here, two were best described as some form of coastal gravity current with associated shock coastal wave (July, 1982 and May, 1985, Appendix 2), while the other propagated as a solitary Kelvin wave (May, 1982). In each case, good 108 agreement was obtained between the theoretical and observed values of the propagation speed, the width of the disturbance and the time and space scales of the initial marine layer disturbance in the Southern Californian Bight from which the disturbances developed. A pronounced difference in forcing and hence initial marine layer disturbance was observed between the solitary Kelvin wave May, 1982 case and the gravity current cases of July, 1982 and May, 1985. In the former, the synoptic forcing was a small closed low which acted much like an impulse wave-maker in a laboratory tank, forcing a localised and substantial initial disturbance in the Bight. For the gravity current cases on the other hand, the synoptic forcing consisted of a much larger feature (either an open baroclinic trough or a large cut-off low) which caused a smaller increase in the marine layer depth (than the May, 1982 case) in the Bight but over a much larger area. This large area of raised marine layer then acted as a reservoir for propagation of the disturbance northwards as a coastally trapped gravity current. Other substantial differences between the solitary Kelvin wave May, 1982 case and the gravity current July, 1982 and May, 1985 cases involved the nature of the propagation. In the solitary Kelvin wave event, propagation was steadily progressive and resulted in short term hump-shaped displacements of the surface pressure fields. Wind shifts during this event were short term too. For the gravity current cases, propagation was unsteady and spurting and the perturbations of the marine layer winds and pressure fields were more variable and of longer duration. The return of these fields to their pre-disturbance values was also irregular and completely unlike the pulse-like character of the solitary Kelvin wave case. A final difference concerned the way in which the disturbances ceased propagation. In the May, 1982 case, it was shown that it was the large convex bend in the coastal mountains at Cape Mendocino which prevented further propagation of the solitary Kelvin wave despite 109 the fact that the displacements of the winds and pressures in the marine layer just south of the Cape were substantial and indicative of a still energetic disturbance. Thus, topographic effects were seen to be crucial for determining the lifespan of this disturbance. On the other hand, the cessation of the gravity current events was fundamentally determined by the strength of the reservoir in the Southern Californian Bight. When this reservoir became too weak to overcome the dissipative effects of the opposing northerly flow and gaps and bends in the topography then propagation ceased. In the July, 1982 case, the disturbance reached Cape Blanco in southern Oregon before this happened whereas in the May, 1985 example, propagation as far as the northwestern tip of Vancouver Island was observed. Finally, it is considered here that the analysis presented in this Chapter has contributed towards resolving the vigorous dispute in the literature over the interpretation of the May, 1982 case as a solitary Kelvin wave (Dorman, 1985, 1988) or as a gravity current (Mass and Albright, 1987, 1988) in favour of the former. It is also felt that an improved dynamical understanding, in terms of the modified Kubokawa and Hanawa model developed in Chapter 2, of the gravity current events of July, 1982 and May, 1985 has been achieved. 110 Chapter 4. The Southern African Coastal Low 4.1. Introduction The coastal low of Southern Africa is a shallow, mesoscale low pressure disturbance that propagates eastwards around the subcontinent and whose distinct features arise from the interaction between the atmospheric stratification, the synoptic weather patterns and the topography of the region. Due to the frequency of its occurrence, which may be taken as about once every six days (Preston-Whyte and Tyson, 1973; CLW, 1984; Kamstra, 1987), and the fact that the intensity of the disturbance is greatest near the coast and decays seawards (Gill, 1977; Anh and Gill, 1981), the coastal low is a dominant feature in the weather of this area. As a result of its prominence, the coastal low is well known in the local weather lore and was also the first coastally trapped disturbance to be modelled analytically (Gill, 1977). As illustrated in Chapter 2, the coastal low is characterised in its formation stages by insignificant nonlinearities and by blocking sufficiently weak that the interaction of the incident synoptic flow with the coastal escarpment is well described by quasi-geostrophic dynamics. This type of dynamics, which is amenable to analytic treatment, was assumed by Anh and Gill (1981) and Bannon (1981) in their models of coastal low generation. Note that although the blocking is weak enough for the overall flow to fall within the quasi-geostrophic regime, it is still sufficiently strong to cause the low-level flow to be diverted along the coastal escarpment. Also shown in Chapter 2 was the fact that, because of its small R/L ratio and offshore Rossby number Rx, the coastal low was likely to exhibit Kelvin wave characteristics rather than those of the nonlinear gravity current. Strong evidence for I l l the validity of the Kelvin wave model of the coastal low will be found in the data analysed in this chapter. In all observed cases, the propagation of the coastal low is preceded by the eastward movement of a migratory subtropical anticyclone and followed by a mid-latitude frontal system of the South Atlantic Ocean. Acting as an obstacle to these westerly perturbations are the coastal mountains of the region which rise steeply to a height of over 1 km (see Fig. 4.1). With the exception of a few river valleys, the coastal mountains form a relatively smooth, semi-circular barrier so that with the interior plateau, Southern Africa resembles a dome with a horizontal radius of curvature of about 900 km. This steeply sloping, curved surface is able to steer low level coastal weather efficiently around its periphery (Taljaard, 1972). Due to the presence of a strong subsidence inversion at about the height of the coastal mountains, flow constrained in the horizontal by the topography is trapped in the vertical. Climatologically, this inversion is typically 750 - 1200 m in height and 3 - 5 °C in strength (Preston-Whyte et al., 1976). Trapping in the horizontal occurs via the effects of Coriolis forces on the alongshore flow which, below the inversion, may be assumed to be in geostrophic balance with the across-shore pressure gradient, Gill (1977). The low level across-shore flow, which is usually not geostrophic, must vanish at the coastal mountains due to the barrier effect and is generally negligible within a coastal zone about a Rossby radius seawards of these mountains (Gill, 1977; Anh and Gill, 1981; Bannon, 1981; CLW, 1984). As a result, the initial flow is constrained to move eastwards around the coastal mountains within a narrow zone of the order of the Rossby radius (100 - 300 km) in width. Gill (1977) first suggested that the structure and propagation features of the coastal lows are similar to that of coastally trapped waves in the ocean. Through use of the 112 Figure 4.1. Topographic and location map of Southern Africa. AB refers to Alexander Bay, CT to Cape Town, CA to Cape Agulhas, PE to Port Elizabeth, and D to Durban. The first contour inland from the coastline is the 500 m contour, that enclosing the stippled region is the 1000 m contour, and that enclosing the black area is the 1500 m contour. 113 nonlinear shallow water equations on the f-plane, with a reduced gravity model of the stratification, Gill showed how interaction of incident barotropic westerly waves with the escarpment generated coastal lows in the form of a Kelvin wave. The nonlinear advection terms were shown to cause steepening of the wave front, hence inducing the rapid changes in wind speeds and direction and in inversion height which sometimes occur during coastal low passage. Following on from Gill's study, Bannon (1981) and Anh and Gill (1981) presented models that examined the interaction of the incident synoptic waves with the escarpment in more detail and also gave more information on the dispersion properties of the thereby generated Kelvin waves. To accomplish this, a two layer model of the atmospheric stratification on the beta-plane was employed, and to allow an analytic solution, the nonlinear terms were neglected. Despite the seemingly restrictive assumptions made, the analytical models were able to explain most of the observed features of coastal lows and are consistent with the results obtained from a numerical forecast model (CLW, 1984). One of the features that the models have not explained entirely is the fact that one tends to observe predominantly coastal lows rather than highs. A linear Kelvin wave model is unable to account for this feature so the preferential excitement of coastal lows was attributed to nonlinear effects by Gill (1977) and Anh and Gill (1981). While not disputing this mechanism, Bannon (1981) also considered the effects of the forcing anticyclone and offshore flow beneath the inversion to be important. In this chapter, new data and a model will be presented that show the importance of this offshore flow in leading to the formation of a coastal low as opposed to a high. It will be argued that since this flow is warmer and less dense after descending the escarpment than the maritime air it replaces, it represents a vertical advection of 114 buoyancy. This buoyancy advection will be shown to cause both a lowering of inversion height and to act as a source of cyclonic vorticity, and hence the offshore flow gives rise to a low pressure cell. In agreement with the theory of Gill (1977) and Bannon (1981), the coastal low is considered here to be some form of Kelvin wave and the presented data are shown to be consistent with this hypothesis. Additionally, this chapter will make-two new contributions to the understanding of coastal low dynamics. Firstly, the theory of Section 2.4 will be applied to the South African coastal atmosphere to show why the disturbance propagates as a mesoscale low rather than as a high. Then, for the first time, the time-height variability of the lower atmosphere during each coastal low event will be examined in detail as the system propagates eastwards around the coast. These data, which were obtained in raw form from the South African Weather Bureau, consist of atmospheric soundings to a height of 700 mb taken at the four coastal stations of Alexander Bay, Cape Town, Port Elizabeth and Durban (see Fig. 4.1 for locations) during a 36 hour period both preceding and following the arrival time of the low at a given station. To illustrate that the coastal low is a year round phenomenon, events are chosen from the months of February, April and September. The selected events are all considered representative of coastal low behaviour according to the guidelines developed by the community of synoptic meteorologists and oceanographers in South Africa (CLW, 1984). Since the behaviour of the coastal low is fundamentally the same in each event, the 3 cases are not considered separately. Instead, the synoptic pressure distributions (Figs. 4.2a-c), time-height variability of wind speed and direction (Figs. 4.3a-c), temperature and dewpoint profiles (Figs.4.4a-c) and geopotential height anomalies (Figs. 4.5a-c) for each event at the 4 stations will be 115 considered together and explanations of the minor variations between the events ~ given concurrently. 4.2 Synoptic Conditions Figures 4.2a-c are the above mentioned synoptic pressure maps which show that in all three cases coastal low formation near 25 S on the west coast of Southern Africa was preceded by the ridging of the South Atlantic Anticyclone to the south of the landmass. Note that in each case, the arrow indicates the position of the coastal low. This ridging and the trailing cyclone development to the southwest over the South Atlantic Ocean (typically 40-50 S,0-15 E and see the third panel from the left of Figs.4.2 a-c) are the surface expressions of an upper level Rossby wave in the general synoptic cycle of the southern midlatitudes. Resulting from the ridging are easterly winds over the interior which, on descending the escarpment, cause offshore flow along the west coast initially. Figures 4.3a-c indicate the wind profiles measured at the four coastal stations (Alexander Bay, Cape Town, Port Elizabeth and Durban), showing that this offshore flow generally occurs all along the South African coast at a height above the inversion layer. As will be examined later, the offshore flow provides the forcing for coastal low propagation. As the ridging process continues eastwards, offshore flow extends further south on the west coast and hence coastal low propagation is induced. Meanwhile, far to the southwest the trailing frontal system is beginning to develop (third panel of Figs.4.2a-c). Eventually, the ridging anticyclone pinches off to form a migratory high which, as it tracks eastwards, induces offshore flow along the south and east coasts and, in tandem, the coastal low to propagate around Cape Agulhas (southeast of Figure 4.2. Synoptic pressure maps Tor the surface at 14 00 UTC daily obtained from the South African Weather Bureau. The map borders for each panel are 20-45 S, 5-45 E. In each figure, H refers to the ridging anticyclone, the bold line to the trailing midlatitude frontal system and the arrow to the coastal low. Case 1 extends over 6-11 February, 1981; Case 2 over 15-20 September, 1985 and Case 3 over 18-23 April, 1980. Pressures over the sea are given as the difference in millibars from 1000 mb and the isobars are contoured at 2 mb intervals. Over the continent, the isolines refer to the 850 mb surface contoured at 10 gpm intervals. 117 Case 3 -24 -12 o n ; 2 24 h -24 -12 0 12 *24 h * -24 -12 Figure 4.3. Time-height plots (4.3a refers to Case 1. 4.3b to Case 2 and 4.3c to Case 3) of wind speed (contoured at 4 m/s interval) and direction (arrows, where the point of the arrow refers to the time-height co-ordinates of the datum). Wind speeds greater than 8 m/s are stippled. The dashed line refers to the minimum wind speeds recorded during each event. On the time axis, 0 refers to the time of coastal low arrival, 12 to 12 hours after arrival and -12 to 12 hours before arrival, etc. On the vertical axis, the pressures are in millibars and the labels A B , CT, etc., are as in Fig. 4.1. 118 Cape Town at 35 S,20 E) and along the south and finally, east coasts. Since the trailing cyclones are faster than the coastal low (compare separations between the low and the cyclone in panels 3-6 of Figs. 4.2a-c), the distance between the two systems decreases as they move eastwards. In Cases 2 and 3 (panels 5 and 6, Figs. 4.2b-c), the coastal low is overtaken and dissipated within the trailing front associated with the mid-latitude cyclone whereas for Case 1 (panel 6, Fig. 4.2a), the low dissipates on the east coast north of Durban before the front can catch up with it. The latter situation arises due to the more southerly track of the midlatitude cyclone in Case 1 and is more typical of summer conditions when the subtropical high pressure belt lies nearer the pole. The speed of propagation of the coastal low on the west coast is largely dependent on the ability of the ridging anticyclone to lower and strengthen the pre-existent inversion. Indeed, in cases where this ridging process is blocked by other synoptic scale systems, the coastal low is often observed to stagnate and eventually, decay (CLW, 1984). However, in all the examples discussed here, the coastal low is observed (panels 1-3, Figs. 4.2a-c) to move southwards along the west coast at a rate of 3-5 degrees of latitude per day so that about a day (a more accurate time of arrival and hence speed will be determined from the sounding data in Section 4.3) after formation near 25 S the low has reached Alexander Bay (28 S, 16 E). By the following day, the coastal low is seen to approach Cape Town (34 S, 18 E) after which the system rounds Cape Agulhas to propagate along the south coast. On the south and east coasts, both the speed of propagation and the lifespan of the system are influenced by the intensity and proximity of the trailing midlatitude cyclone. Where the latter is either relatively weak or lies far to the south of the landmass, which is usually the case except in winter, the low is able to propagate more quickly than on the west coast. Thus, on the south coast, the low moves 7-9 degrees of 119 longitude to a position near Port Elizabeth (34 S, 26 E) in the day following its arrival at Cape Town (panels 3 and 4, Figs. 4.2a-c). Speeds on the east coast near Durban are generally slower due to a weaker inversion and hence there is less effective trapping of the low in the vertical. This weakening of the inversion probably results from the greater surface heat fluxes over the warm sea surface temperatures of the Agulhas Current, which skirts the coast from Durban to Port Elizabeth before moving offshore. Within a day or less of arriving at Port Elizabeth, the low is seen (panels 4 and 5, Figs. 4.2a-c) to reach Durban (30 S, 31 E) after which the system moves out to sea and dissipates (Case 1 - panel 6, Fig. 4.2a) or is overtaken by the trailing frontal system (Cases 2 and 3 - panel 6, Figs. 4.2b-c). Since the coastal mountains turn significantly inland after Durban, the ability of the topography to steer the low in the horizontal is quickly lost so that together with the weaker subsidence inversion over the warm Agulhas waters, there is little effective trapping on the northeastern coast of Southern Africa. Additionally, the offshore wind forcing of the coastal low is generally weaker near Durban. North of Durban, conditions which lead to this offshore wind forcing are known to be rare (CLW, 1984). Surface friction and the synoptic scale flow are then able to dissipate the mesoscale coastal low rapidly. For example, Figs. 4.2a-c (panel 6) show that the system disappears in a day or less. Case 3 (Fig. 4.2c) illustrates particularly well the close association of the coastal low with the synoptic forcing. As the trailing cyclone approaches the coastal low near Durban, the synoptic cycle is already causing the next ridging of the South Atlantic Anticyclone west of Cape Town (panels 5 and 6) and hence the formation of a new coastal low off the northwestern coast of Southern Africa near 20 S, 13 E. This system was observed (on synoptic maps not shown here) to propagate vigorously 120 along the west coast under the influence of strong synoptic forcing so that within a day it was located near 28 S, 15 E, and, the following day, the low was observed on the south coast at Port Elizabeth. In summary, Fig. 4.2 has shown that the eastwards propagation of the coastal low around the Southern African coast is very closely tied to the synoptic weather cycle of the southern subtropics/mid-latitudes. It is this close association which provides continual forcing for the coastal low and, as the analysis below will show, enables the coastal low to propagate in a coherent wavelike manner. 4.3 Atmospheric Sounding Data Figs. 4.4a-c show temperature and dewpoint temperature profiles through the lower atmosphere as derived from soundings taken every 12 hours by the South African Weather Bureau at the four coastal stations already mentioned. In virtually every case, a pronounced temperature inversion (indicated by an arrow in Figs. 4.4a-c) of the order of 5 - 10 °C is seen to exist. Note that in some cases, pronounced heating at the surface prevents the formation of a temperature inversion. In these cases, the inversion layer can still be detected by the substantial decrease in dewpoint temperature that always occurs over this layer (CLW, 1984) and which marks the transition from a moister maritime air mass below to the dry continental air mass above the inversion. As the coastal low approaches a given station, the inversion is observed to lower and strengthen and then increase in height and weaken after coastal low passage. Generally, the actual minimum in inversion height occurs at the time of passage, behaviour which is consistent with a linear Kelvin wave. Apart from the drop in 121 Figure 4.4. Temperature (right hand curve) and dewpoint temperature (left hand curve) profiles through the lower atmosphere. Fig. 4.4a refers to Case 1, 4.4b to Case 2 and 4.4c to Case 3. On the vertical axis, the pressure level p in the atmosphere is given as 1020 - p millibars. The height of the inversion base is indicated by the arrows. On the vertical axis, the labels AB, CT, etc. are as in Fig. -4.1. 122 H U M t (US) ( W O M B - K) g 8 g 3 i 8 MOGMT ( M O ( t a n a - r) g § g 8 8 8 MDCMT (W» (tCOOMi - I) • o g i i 8 8 8 KTCHT (MB) ( l a O M t - I) . » g § g 3 8 8 *r MOCHT (Ma) (1S39M1 - I> . g i i 8 3 8 Figure 4.4a . x i i 5 8 5 . » a j 8 8 8 g g i 8 j 8 • V O * (M) (I83QW - II . « 8 8 8 8 8 Figure 4.4b 124 8 8 g g g g m o i t pa) u n a - q . i g g g g g a 8 e V ?• / 1 V 5 /1 e 8 — — 3 • « 0 « t {«•> ( 1 0 3 M • I) g g i g g 8 g § j g 2 8 o g 8 g g g g r 0«> ( O T M - n g g g 8 2 8 5 s 5 a" ~ I ^ s 5 1" 2 Is a C 8 8 8 8 8 S 8 2 g r 8. Figure 4.4c 125 inversion height, coastal low passage is also characterised by variations in the dew point temperature, a shift in wind speed and direction and most reliably, a minimum in the geopotential height of the various pressure levels below the inversion. From Figs. 4.3a-c, the coastal low is. generally preceded by offshore flow which switches to an on- or alongshore direction as the system passes through. Offshore flow on the west coast (Alexander Bay and Cape Town) refers to an easterly wind whereas on the south (Port Elizabeth) coast it is northwesterly or northerly and on the east (Durban) coast it is westerly or northerly (compare directions with coastline orientation in Fig. 4.1). Note that the offshore flow at Cape Town, Port Elizabeth and Durban is often located in and above the inversion due to the strength of the near-surface easterly trade winds along the south and east coasts. Since this offshore flow is warm and dry, being of continental tropical origin, near-surface temperatures prior to coastal low arrival rise reaching a maximum when the inversion drops to its lowest point (Figs.4.4a-c). At this point, the extent of mixed layer that is subject to surface heating is shallowest, hence the higher temperatures. The onshore flow behind the trailing edge of the coastal low is of maritime origin and hence represents a cooler and moister airmass. As a result, near-surface temperatures tend to decrease substantially after the passage of the coastal low and there is usually a marked rise in the dew point. The latter leads to a significant increase in cloud cover below the inversion, invariably as stratus. In the case of Alexander Bay, this onshore flow often does not reach the surface due to the strong stratification present over the cold Benguela Current. As the trailing edge passes through, surface pressure and the inversion height increase and the near-surface winds shift to alongshore and strengthen. Thus, at Alexander Bay the direction is northerly or northwesterly, at Cape Town 126 northwesterly, at Port Elizabeth westerly or southwesterly and at Durban southwesterly or southsouthwesterly (Figs. 4.3a-c). Associated with the rising inversion is an increase in the near surface moisture content. Typically, dew points near the inversion (900 mb) rise by about 8 °C following coastal low passage. Figs. 4.3a-c show that the wind shift during coastal low passage tends to be sharper on the south and east coasts than on the west coast. This phenomenon is well known locally, giving rise to the term Southwest Buster, and arises partly through advection by the closely approaching frontal system. Boundary layer heat fluxes, caused by variations in sea surface temperatures, also play a role by stabilising the marine atmosphere on the Benguela-facing west coast, thus preventing buster formation there, while on the warmer Agulhas-facing south and east coasts, momentum is more readily transferred to the surface. Other local variations in coastal low weather arise through the effects of topography or surface heating. For example, on the west coast the wind shift near the surface may be obscured to some extent where local topographic influences are able to cause sheltering effects. Where offshore data are available, the wind shift along this coast is most clearly observed 10-20 km offshore or at protruding headlands (CLW, 1984). Such topographic effects are particularly common near Cape Town (e.g., Case 2, Figs. 4.3a-c and see Jury, 1987) where some of the prominent mountains lie perpendicular to the coastline and therefore modify the local windfield. Sometimes, for example Case 2 at Port Elizabeth (Fig. 4.3b) a strong nocturnal surface inversion may prevent the wind shift being observed at the surface at the time of coastal low passage. In other cases, the lag between the inversion height minimum and the wind shift may be due to nonlinear steepening of the coastal low (see Bennett (1973) arid Gill (1977)). 127 Another common feature is the generally higher inversion level on the south and east coasts than on the west coast. This feature is a result of the greater surface heating on the Indian Ocean side of Southern Africa with its warm Agulhas Current. Sea surface temperatures on the Indian Ocean coast are typically 10 °C warmer than those on the cold Benguela Current South Atlantic side (CLW, 1984; Shannon, 1985) where coastal upwelling often reduces the inshore temperature still further. Finally, consider the differences in geopotential height from the mean that are observed during coastal low passage through a given station (Figs. 4.5a-c). Since the arrival of the coastal low at a given station is associated with a minimum there in the geopotential height at the surface, cases which are free from any upper level influence should show a core of negative values at levels below the inversion clustered around the time of arrival. This distribution of geopotential differences is indeed observed in all cases excepting Case 1 and 2 at Cape Town (Figs. 4.5a and b), Case 2 at Port Elizabeth (Fig. 4.5b) and Case 3 at Durban (Fig. 4.5c). These exceptions may be treated in two different categories. In those examples (Cases 1 and 2 at Cape Town, Case 2 at Port Elizabeth) where there is some interference from the upper level Rossby wave, the core of negative values extends to levels above the inversion (i.e., above about 850 - 900 mb). On the other hand, if upper level cyclogenesis occurs (Case 3 at Durban), then the extreme negative values are found above the inversion. It is considered that the most reliable signature of coastal low passage at a given station is the surface minimum in geopotential height. Using this indicator, the time of passage for each event is determined and hence the speed of propagation for each case can be calculated (see Table 4.1). In the next section, this value is compared with the theoretical phase speed of a linear Kelvin wave. 128 CO < 800 900 1000 r -U 800-1 900 \ I \ I • 1^5* Case 1 AO / t _IZ L / / 1000 k «/// /// JsilA f f I / / 7\ f "30 / / / i ! o 800 \ 15 900 V 1000 Q 800 900 1000 I \ l l 0/ k 7 1 If - J 11/15 \ \ \ I iifhl v\t so' , ; : / / i ' /-I /-15 IS I Is I -1 I I io I I -24 -12 0 12 24 h b Case 2 J I / / / / / / / 11 / -15 I I /// i i / / 39} \ \ i M 1 \ \ ) ) I If/-/ / / / / , »» / / ill* 7 t .1 - I 151 I [ \( WW H I -24 -12 1 •11 I / | l he 0 12 24 h c Case 3 30 / / / , ' ML \ \ i / l»-15 M l / ' \ / / /I Mil.Ik Ski/T - / » //--24-12 0 12 24 h Figure 4.5. Time-height variability of the geopotential anomalies (gpm) from the mean recorded during the event. Fig. 4.5a refers to Case 1, 4.5b to Case 2 and 4.5c to Case 3. Anomalies below -30 gpm are shaded. Labels on the horizontal axis are as in Fig. 4.3. On the vertical axis, pressures are in millibars and the labels AB, CT etc. are as in Fig. 4.1. 129 4.4 Comparison of coastal low behaviour with Kelvin waves The consistency of coastal low behaviour with Kelvin wave dynamics is now considered. Thus, the coastal low is assumed to be an internal Kelvin wave propagating in the inversion layer with phase speed (g'H)-'-/2. From the inversion data obtained from the South African Weather Bureau and plotted in Figs. 4.4a-c, the Kelvin wave phase speed is calculated (Table 4.1). The observed speeds were obtained using the time of arrival at a given station as determined from the sounding data (Figs. 4.4-4.5a-c) and extended versions of the synoptic maps (Figs. 4.2a-c) and reflect the average observed propagation speed of the coastal low. In each case, good agreement between the observed and theoretical speeds is seen to exist. Maximum probable errors in the observed speed data are estimated from the maximum error range in the observation of coastal low arrival (±6 hours) while those for the computed speeds are estimated from the maximum error range in the potential temperature (±0.5 K) and inversion height (± 50 m). At Durban however, the theoretical values are not considered completely reliable for several reasons. Only in Case 1 (Fig. 4.2a) is the coastal low still well separated from the trailing frontal system. In the other cases, the linear dynamics is probably violated by advective forcing from the nearby front and so (4.1) is no longer valid. In addition, the generally higher and weaker inversion at Durban makes for less effective trapping of a Kelvin wave feature and this is enhanced by the turning inland of the topography downstream to the north of this station. Additional support for the Kelvin wave hypothesis arises from the coastal low behaviour as displayed by the sounding data of Figs. 4.3-4.4a-c. The relatively 130 T a b l e 4.1 Observed p r o p a g a t i o n speed and computed speeds f o r t h e S o u t h e r n A f r i c a n c o a s t a l low. Case Observed Speed LKW Speed G-C Speed SKW Speed 1 A.B. 7 ( ± 2 ) 7 9 9 C T . 9 ( ± 3 ) 8 1 6 1 0 P.E. 9 ( ± 3 ) 9 1 8 1 1 D. 5 ( ± 1 ) 3 1 1 4 2 A.B. 6 ( ± 2 ) 6 1 0 n 1 C T . 8 ( ± 2 ) 7 1 3 9 P.E. 9 ± 3 ) 9 1 9 1 1 D . 8 ± 2 ) 8 1 5 9 3 A.B. 9 ± 3 ) 1 0 1 5 1 3 C T . 1 0 ± 3 ) 8 1 2 1 0 P.E. 8 ± 2 ) 8 1 3 1 0 D. 3 < ± 1 ) 3 •9 4 Notes : 1 . A l l speeds are i n m/s 2 . The e s t i m a t e d maximum probab l e e r r o r i n the computed speeds i s 1 m/s f o r LKW and SKW, 2 m/s f o r G-C 3 . LKW r e f e r s to l i n e a r K e l v i n wave, G-C to g r a v i t y c u r r e n t and SKW to s o l i t a r y K e l v i n wave. 131 coherent, variations in the inversion layer height and wind direction are characteristic of Kelvin wave dynamics as opposed to the gravity current hypothesis advanced for the coastally trapped disturbances off the west coast of North America (Mass and Albright, 1987; Dorman, 1987). Since gravity currents tend to propagate in an unsteady, surging fashion (Simpson, 1987), they typically cause abrupt changes (and often spurts) in the observed winds and inversion height (see Dorman, 1987). This sort of propagation is not observed in the coastal low cases examined here. As discussed in Chapters 2 and 3, the existence of a density front between the gravity current and the ambient atmosphere leads to similar abrupt changes in the density and temperature of the marine layer (including the surface air) as the leading edge of the gravity current passes through a given station. These abrupt changes have not been observed during coastal low passage for those cases where hourly resolution data is available (CLW, 1984; Heydenrych, 1987; Hunter, 1987). Perhaps the most telling features of the coastal low that preclude this phenomenon from being classified as a gravity current, are the lowering in the inversion height and geopotential (and hence pressure) with propagation and the way in which the coastal low is generated. The coastal gravity currents of North America and Australia all form through the passage of some synoptic scale system across the coastal mountains leading to an area of raised marine layer in a bight (Southern Californian or Great Australian) region which then acts as a reservoir (as indicated by widespread stratus formation) for gravity current propagation. These reservoirs are clearly visible on satellite imagery (Holland and Leslie, 1986; Dorman, 1987; Mass and Albright, 1987; see Figs. 3.15a-b, 5.2). No such situation occurs for the coastal low, which as Section 4.5 will show, forms through vertical advection of buoyancy via the warm offshore winds off the interior escarpment. Indeed, examination of satellite imagery for the coastal low cases 1-3 as well as several others, showed no reservoir formation on the 132 west coast of Southern Africa. As stated above, the gravity current events of North America and Australia are always associated with a rise in the inversion height and an increase in surface pressure (e.g., Figs. 3.8a, 5.7, A5, A12-13, A15-17) as the denser, cooler air of the gravity current replaces the ambient air. On the other hand, the coastal low causes a decrease in surface pressure (Figs. 4.2, 4.5a-c) and a lowering of the inversion (Fig. 4.4a-c). Lastly, Table 4.1 shows that the empirical propagation speed (2.11) of an atmospheric gravity current is too fast to explain coastal low propagation. Note that the surface pressure rise 6P was obtained from the surface geopotential data (which is contoured in Figs. 4.5a-c) and extended versions of the synoptic maps (Fig. 4.2) given here. The solitary Kelvin wave hypothesis, which has been suggested by Dorman (1985) as explanation for the Californian event of May 4-7, 1982 (see Chapter 3), is not considered likely here either for the following reasons. Firstly, the observed coastal low speeds are more accurately described by a linear Kelvin wave (Table 4.1). Secondly, the coastal low is observed in all cases to propagate without loss of identity around the pronounced bend in the coastal mountains just east of Cape Town. Such behaviour is consistent with that of linear Kelvin waves (Miles, 1972; LeBlond and Mysak, 1978) but not with solitary Kelvin waves (Miles, 1977; Reason and Steyn, 1988). Finally, note that a solitary wave requires dispersive effects to balance the nonlinearities. While it is likely that the mean wind during the May, 1982 Californian event (see Chapter 3) was able to provide the necessary dispersion, there is no obvious dispersive mechanism in the coastal low case. 133 Finally, note that the computed Rossby radius during the events examined here (100-150 km), is consistent with the observed width scale of the coastal low as determined from the synoptic pressure maps (Figs. 4.2a-c). Hence, the fundamental length scale requirement for a Kelvin wave is satisfied by the coastal low observations. In summary, the linear Kelvin wave model of the coastal low is felt to be most appropriate. Based on this viewpoint, the forcing of the coastal low is now considered. 4.5 The Contribution of Berg Winds to Coastal Low Forcing In this section, the contribution of berg wind flow off the interior plateau of South Africa to the forcing of the coastal low will be examined. For the purposes of this discussion, a berg wind (a local term) is defined as a flow originating from the continental tropical airmass found over the interior plateau which, after descending the escarpment, reaches the low-lying coastal areas as a warm, offshore breeze. Similar flows are found in other parts of the world, e.g., the Santa Ana wind of California. It must be emphasized here that the berg wind should not be confused with the offshore flow that often occurs near the surface immediately ahead of the leading edge of the coastal low. Berg winds, with their origin over the interior plateau must have upper layer support (i.e., exist at and above the inversion level - see Figs. 4.3a-c) and not necessarily, at the surface. On the other hand, the surface offshore flow is part of the mesoscale cyclonic circulation of the coastal low itself and as such is confined below the inversion. Since the temperature difference between the adiabatically warmed berg wind airmass and the maritime tropical air usually present at the coast may be 10 °C or more, the berg flow represents a considerable buoyancy input into the marine atmosphere during which dense air is replaced by less dense. As a result, pressures 134 tend to fall along the coast throughout berg wind events. The extent of this input (i.e., the temperature difference) is often observed some distance offshore, e.g., well over 20 km seawards of the Cape South Coast (Hunter, 1987). The mechanism by which the berg wind is induced to flow off the interior plateau invariably involves the interaction of some synoptic weather system with the escarpment. Thus, on the west coast of South Africa a berg wind usually results from the ridging of the eastward side of the South Atlantic Anticyclone along the south coast whereas on the southern and eastern coasts, berg winds generally precede the eastward propagation of synoptic troughs across the southern half of the country. In many cases, the ridging anticyclone along the south coast buds off to form a migratory high pressure cell which as it tracks southeastwards may induce berg wind conditions along the south and less commonly, the east coasts. This migratory high is observed in all three cases studied here (see Section 4.2) and is responsible for the observed berg winds all around the Southern African coast (see Section 4.3). Since these synoptic conditions are the very same as those under which coastal low propagation may occur, it is argued here that the resulting berg winds are the local expression of the overall interaction of this synoptic forcing with the escarpment. Although the existence of the berg winds is in theory (Gill, 1977; Anh and Gill, 1981) unnecessary for the generation of Kelvin waves in the coastal atmosphere, it will be shown here that these winds are a requisite for the preferential excitement of these Kelvin waves in the form of coastal lows rather than as coastal highs. Also, in all the cases described in section 4.3, berg winds were found to occur a day or so prior to the arrival of the low at a given station (this is true of coastal lows in general, CLW (1984)) and so it is clear that the vertical buoyancy advection associated with berg winds must play a significant role in the continued forcing of the coastal low. 135 Additional to the effect of the buoyancy input, the berg wind also acts as a source of cyclonic vorticity due to vortex stretching as it descends the escarpment. In general terms, the anticyclonic vorticity present over mountainous regions tends to decay due to boundary layer.friction (Smith, 1979). However, this decay represents an increase of potential vorticity so that when the fluid columns leave the mountain, their increased potential vorticity is converted to cyclonic relative vorticity so that as a whole, potential vorticity is conserved. Thus, as shown explicitly in Section 2.4, a subsiding berg wind will lead to cyclonic vorticity at the coast which, together with the falling pressures resulting from the lower density of this wind, will give rise to a coastal low that is then trapped to within a Rossby radius of the escarpment through Coriolis effects. Since berg winds are so commonly associated with coastal low propagation, it would seem that the preceding vorticity argument offers a more satisfying explanation for the fact that a low as opposed to a high is formed than the heuristic nonlinear argument of Gill (1977) in which coastal lows were viewed as being likely to intensify since the local winds would be towards the system whereas a coastal high would tend to dissipate as a result of their outflowing winds. Indeed, without identifying the berg wind as the responsible agent, Bannon (1981), in his model of coastal low dynamics, stated that divergent (i.e. offshore) flow below the inversion should give rise to a coastal low whereas onshore flow would produce a coastal high. Although the importance of berg wind forcing in the dynamics of coastal low propagation has been recognised by synoptic meteorologists in South Africa (CLW, 1984), its contribution has not been explicitly modelled. With this in mind, the simple linear model of the effects on the inversion height of a subsiding wind developed in Section 2.4 can be applied. Note that the initial stages of the coastal low are typified by a small Rossby number (winds being less than 5 m/s and offshore 136 scale of 200-300 km at 30 S) and a subcritical Froude number (these winds are less than the Kelvin wave speeds in Table 4.1) and so a linear model is valid. In this model, it was shown that a subsiding wind acts to cause a relative decrease in the inversion height (equation 2.16) and forces the propagation of a linear Kelvin wave on the inversion. This decrease in the inversion together with the generated cyclonic vorticity shows that the subsiding berg winds cause a coastal low rather than a high to form. Following on from this, the effects of periodic forcing were considered. Motivation for considering periodic forcing is that a distinct 6 day cycle is observed (Preston-Whyte and Tyson, 1973; Kamstra, 1987) in the surface pressure spectra measured at coastal South African sites and that, as shown in Fig. 4.3, the coastal low was preceded by berg winds 24 hours prior to its arrival at a given station. The 6 day cycle has been identified with coastal low forcing by Gill (1977) and Bannon (1981). To see the magnitude of the effect of the berg wind forcing, typical values of the parameters are substituted in the solution (2.24-2.25) to the model and the resulting inversion displacement displayed graphically in Fig. 4.6. These are : initial inversion height at coast H 0 = 1000m, R = 137 km, subsiding berg wind W Q = -0.05 m/s, f =-7.29x10"^  s"-*-, Q = 1.2x10"^  s'^ corresponding to a period of 6 days, and offshore extent of forcing a = 50 km. This value for a is chosen because the effects of berg wind events have been regularly observed 20-30 km offshore of the south coast (Hunter, 1987) and, occasionally, as far as 100 km offshore (Shannon, 1985). Then, (2.24-2.25) yield a decrease in the inversion height of about 20% greater than the free wave solution (Fig. 4.6) and hence the effect of berg wind forcing on the inversion displacement, as revealed by this rather crude model, is substantial. 137 HEIGHT OF INVERSION (M) Figure 4.6. The free (unforced) linear Kelvin wave (upper curve) and berg wind forced linear Kelvin wave (lower curve) model solutions. 138 4.6 Conclusion In this chapter, atmospheric sounding data and synoptic weather maps have been presented which show that the coastal low is a shallow mesoscale disturbance that propagates coherently in an eastwards direction around the coastal mountains. The energy of the coastal low is trapped in the vertical by the strong subsidence inversion that is typically situated at a height of 1 km, just below the crests of the mountains, and which is a climatological feature of the region. In the horizontal, the escarpment traps energy to within a Rossby radius through Coriolis effects on the alongshore flow. The hypothesis of Gill (1977) that the coastal low is essentially a coastally trapped Kelvin wave is reinforced by the time-height variability of the lower atmosphere during the three events examined here. It was observed that the displacement of the inversion interface, the surface pressure as measured by the geopotential height at 1000 mb and the shift in the winds following coastal low passage were all consistent with an internal Kelvin wave propagating in the marine layer. In addition, good agreement was found between the observed propagation speed of the coastal low and the theoretical phase speed of a linear Kelvin wave. Forcing of the coastal low occurred via the ridging of an eastward moving synoptic scale anticyclone to the south of the continent. This ridging causes easterly winds over the interior plateau which on descending the escarpment become a warm offshore flow or berg wind. The buoyancy advection. resulting from the berg wind together with the cyclonic vorticity acquired during, the descent of this wind off.the interior escarpment cause a coastal low as opposed to a coastal high to form. Continued migration of the anticyclone eastwards in each observed case resulted in 139 berg wind conditions all around the Southern African coast and hence the coastal low was forced throughout its propagation path, a situation which is typical (CLW, 1984). A simple, linear model of the berg wind forcing was able to reproduce the essential features of the observed behaviour. Trailing the coastal low in each case was a mid-latitude cyclone which, although well behind on the west coast, rapidly approached the coastal low on the south and east coasts. Except for Case 1, which occurred during high summer, the cyclone overtook and dissipated the coastal low before it could reach the bend in the coastal mountains northwest of Durban. This behaviour, which probably results from the extreme southerly displacement of the midlatitude westerlies and hence cyclone tracks during summer, is the only example of potential seasonal variability in coastal low propagation that could be detected in the three cases examined. In all three cases, the coastal low was observed to propagate around the rather sharp convex bend in the coastal mountains near Cape Town. Such behaviour is consistent with what is expected theoretically for both unforced (Miles, 1972) and forced (Clarke, 1977a, 1977b) Kelvin waves. These theories predict that at a convex bend ("cape"), the phase speed of the wave should decrease and the inversion displacement increase whereas at a concave bend ("bay") the reverse should occur. Unfortunately, the resolution of the data is not adequate to determine whether this in fact occurs. Nevertheless, the theory does offer a possible explanation for the observation that the propagation speed of the coastal low is often less at Cape Town than that measured along the Cape south coast several hundred kilometres to the east (CLW, 1984, Hunter, 1987). 140 Other influences on coastal low behaviour include the surface boundary conditions. For example, on the south and east coasts the warmer sea surface temperatures associated with the Agulhas Current provide a significant surface heat flux which causes the inversion there to be higher. As a result, coastal low speeds here tend to be faster than those on the west coast, a feature which may be aided by advection by the closely approaching mid-latitude cyclone. On the west coast, cooler sea surface temperatures, resulting in general from the Benguela Current and inshore due to coastal upwelling, cause a lower inversion there. Upwelling is particularly common during the summer and autumn months when the prevailing surface winds (southerly and southeasterly) are appropriately directed to cause an offshore Ekman flux in the oceanic mixed layer. Hence, it is possible that forcing associated with the thermal contrast between the cold sea and the hot land on the west coast may act to trap the coastal low more efficiently which, would in turn result in higher propagation speeds during upwelling episodes. The observed quicker propagation of the coastal low in Cases 1 (summer) and 3 (autumn) than Case 2 (spring) may perhaps be attributed to upwelling influences. Finally, the potential importance of bottom friction, and in particular its land-sea contrast, on coastal low behaviour may be considered. In cases where the coastal low is not overtaken by the trailing mid-latitude cyclone, the coastal low is usually observed (e.g. Case 1) to decay out to sea northeast of Durban (CLW, 1984). Beyond Durban, the effective trapping of the disturbance is reduced substantially by the weaker inversion present there and by the turning inland of the coastal mountains so that surface frictional effects are soon able to dissipate the coastal low. Surface friction variations may also explain the observed tendency (Van Leeuwen, 1972) for the coastal low to propagate more quickly at night than during the day. At night, when a surface radiation inversion often exists below the main subsidence inversion and the propagating coastal low itself, the effects of bottom friction on the disturbance are minimized and hence propagation speeds may be faster than those during the day. It is considered, therefore, that any future modelling efforts on coastal low propagation should attempt to incorporate bottom friction and surface heat flux effects explicitly. 142 Chapter 5. The Coastal Ridges of Southeast Australia 5.1. Introduction Strong coastal ridging along the coast of southeastern Australia is a common feature of the southern summer, typically occurring 5 or more times per month (Holland and Leslie, 1986). The ridging occurs after the approach of a Southern Ocean anticyclone in the Great Australian Bight (see Fig. 5.1 for place names) towards the south of the continent. From the data of these authors, it is clear that the approaching anticyclone generates an initial disturbance in the marine layer at the eastern end of the Bight (narrow cloud band in Fig. 5.2a). This disturbance then evolves into a mesoscale ridge of high pressure which is able to propagate 2000 km or so along the coastal mountains from Mount Gambier in western Victoria to Brisbane. At Brisbane, the coastal mountains are not only much lower but also bend prominently inland so that together with the weakening of the inversion in these almost tropical latitudes, the waveguide for the ridge is lost and the disturbance is unable to propagate further, instead diffracting out over the sea. Satellite imagery (Fig. 5.2b) taken from Holland and Leslie (1986) illustrates this diffraction clearly. Holland and Leslie (1986) single out the 9-11 November, 1982 event as representative of this coastal ridging and as being easily observed due to a minimum of interference from synoptic scale features. In this chapter, attention will also be confined to this event. The approach will be to take Holland and Leslie's (1986) data and show that it fits the modified Kubokawa and Hanawa model (see Section 2.6) of an intruding coastal gravity current with associated solitary Kelvin waves. Note that this approach is consistent with the viewpoint of Holland and Leslie that coastal ridging events initially form as gravity currents which may later have a superimposed Kelvin 143 Figure 5.1. Topographic and station location map of Australia. Unshaded regions are at 0-300 m elevation, stippled regions at 300-900 m elevation and dark areas at greater than 900 m elevation. Adapted from Holland and Leslie (1986). Figure 5.2. Japanese geostationary satellite visible imagery for - a) 00 UTC November 10, 1982 showing the cold outbreak (signified by the narrow cloud band) at the eastern end of the Great Australian Bight and - b) 00 U T C November 11, 1982 showing the stratiform cloud associated with the propagating coastal ridge on the east coast of Australia. Adapted from Holland and Leslie 11986). 145 wave that propagates through and ahead of the current. In the next section, some of the synoptic details of the event are discussed, while in Section 5.3 the above view of the dynamics is considered in detail. 5.2 Synoptic Conditions during the November, 1982 event According to Holland and Leslie (1986), the approach of the forcing anticyclone in the Great Australian Bight generated a low-level cold air outbreak or initial marine layer disturbance (see Fig. 5.2a) along the south coast by 00 UTC on November 10th. Only 24 hours later, the resulting coastal ridge reached its final point of propagation, namely Brisbane. The synoptic situation is shown in Fig. 5.3. Note that ahead of the forcing anticyclone and generated coastal ridge are a short wave trough and frontal zone. These synoptic features only tracked 300 km or so eastwards while the coastal ridge propagated about 2000 km (Holland and Leslie, 1986). Although this figure does not include the observed wind field, it may be inferred from the pressure distribution that the winds along the south coast at the time of formation of the coastal ridge would be either on- or alongshore. This inference is consistent with Holland and Leslie (1986) who state that the winds prior to the generation of the coastal ridge were westerly or southwesterly on the south coast. Also consistent with this generally on- or alongshore flow, is the generation of the disturbance as a coastal ridge rather than as a trough. As shown in Section 2.4, pre-disturbance on- or alongshore flow and offshore flow leads to the formation of coastal highs (North America) and lows (Southern Africa) respectively. Again similar to the North American and Southern African disturbances already discussed, the propagation of the coastal ridge caused significant local weather 146 Figure 5.3. Mean sea level isobaric analyses for 00 UTC, November 10, 1982 (Fig. 5.3a) and 00 UTC, November 11, 1982 (Fig. 5.3b). The forcing anticyclone in the Great Australian Bight is labelled H. To the east of this anticyclone, over the Tasman Sea, is a short wave trough and frontal zone. Isobars are in 4 millibar intervals. Adapted from Holland and Leslie (1986). 147 changes in the marine layer. For example, the arrival of the ridge caused a wind shift from the prevailing synoptic westerlies to southerly, a 5-10 °C temperature drop and the occurrence of low level stratiform cloud (Holland and Leslie, 1986). Fig-. 5.4a indicates that prior to the arrival of the coastal ridge, the lower atmosphere was characterised by a stably stratified layer below 850 mb, above which a well-mixed almost isothermal layer existed to 750 mb. This situation is attributed by Holland and Leslie (1986) to advection of the deep continental mixed layer eastwards over the Great Dividing Range (of height about 1 km) by the climatological westerlies. Once it reaches the coastal areas, this advected layer overlies a low level stable marine layer which is, according to these authors, maintained by sea breeze activity along the coast. Thus, in the opinion of Holland and Leslie, the stable marine layer (i.e., the waveguide for the disturbance) is formed by sea breezes rather than as a result of large scale subsidence associated with the subtropical high pressure belt. Despite this assertion of Holland and Leslie, it appears that the latter mechanism must also play a role because there seems to be little or no change in the marine layer at night when sea breezes do not usually occur. Whatever the exact contribution of large scale subsidence or sea breezes may be to the atmospheric stratification, a lower stable layer capped by a substantial inversion (potential temperature increase of 10-15 °C) was located below the crests of the coastal mountains (Holland and Leslie, 1986) and therefore a waveguide existed for the coastally trapped ridge to propagate. From Figs. 5.4b-c, it is clear that a pronounced change in the structure of the lower atmosphere was associated with the propagation of the coastal ridge. In brief, the top of the stable layer lifted to about 700mb as the cooler, denser air of the disturbance arrived and then subsided following the passage of the ridge. As shown by Holland 148 23Z 17Z 23Z 23Z 23Z 17Z 23Z 23Z 23Z 172 23Z 2 3 * Figure 5.4. Vertical cross-sections of potential temperature (K) at various stations along the east Australian coast for November 9, 1982 (Fig. 5.4a), November 10, 1982 (Fig. 5.4b) and November 11, 1982 (Fig. 5.4c). The stations used and the observation times are indicated on the abscissa. Adapted from Holland and Leslie (1986). 149 and Leslie (1986), the observed increase in surface pressure is entirely accounted for by the propagation of the coastal ridge. At Williamtown, Fig. 5.5 shows that in addition to these pressure and air mass changes, the wind field switched from light and variable (a local sea breeze circulation) just ahead of the leading edge of the ridge to southerly or alongshore, as expected for a Kelvin wave. Also indicated by this figure, is the fact that the maximum winds alongshore occurred between 1600 UTC November 10 and 00 UTC, November 11. Barograph traces of surface pressure (Fig. 5.6), show that this period of maximum winds coincides with the period of maximum pressure. Such behaviour is, like the wind switch to southerly, also consistent with Kelvin wave theory (Gill, 1977; Holland and Leslie, 1986). Additional evidence for the propagation of a coastally trapped ridge can be seen in Fig. 5.6 by comparing the surface pressure traces for the inland stations, west of the Great Dividing Range, with the coastal stations, east of these mountains. It is clear that the forcing anticyclone only produced a propagating trapped disturbance on the eastern side of the Great Dividing Range, i.e., the ridge propagated with the coastal mountains on the left as required by for a Southern Hemisphere Kelvin wave. Further observations of the coastal ridge are discussed in Holland and Leslie (1986). For the purposes of this chapter, the most important of these observations are that the ridge was trapped within a distance of about 300 km from the coastal mountains, which is consistent with the computed Rossby radius of 327 km (see later), had a total lifespan between generation and cessation of propagation of about 2 days and produced all the changes in local weather expected for this type of coastally trapped disturbance (see Chapters 2 and 3). Thus, for the remainder of this chapter, attention will be devoted towards comparing the observations with the modified Kubokawa and Hanawa model. In addition, an attempt will be made to answer some 150 400 -E UJ cr to CO U J tr a. 500 600 700 800 900 1000 308- 7=^ 3 0 4 = -300-f-—f-296 11 16 00 04 11 16 00 04 11 16 00 04 11 9 10 11 12 DATE Figure 5.5. Time-height variability of winds and potential temperatures (K) at Williamtown for 9-11 November, 1982. On the wind arrows, one full feather indicates 5 m/s. Adapted from Holland and Leslie (1986). 151 Figure 5.6. Barograph traces of surface pressure for various inland and coastal stations in eastern Australia. The pressure traces cover the period 00 UTC, November 9 to 12 UTC, November 11, 1982 and the tick marks are at 10 mb spacing. The unshaded region refers to terrain at 0-600m elevation, the stippled to terrain of 600-1000m and the black region to terrain higher than 1000m. Adapted from Holland and Leslie (1986). 152 of the specific, but as yet unresolved, questions raised by Holland and Leslie (1986). These questions are :-1. Under what conditions do the gravity current and/or the Kelvin wave components of the coastal ridging dominate ? 2. What are the effects of the sharp corners in the coastal mountains at Gabo Island and Brisbane ? 3. What are the effects of other coastal mountain irregularities? 4. What is the contribution of coastal temperature gradients? 5.3 Application of the modified Kubokawa and Hanawa model V As described in Chapters 2 and 3, the modified Kubokawa and Hanawa model for coastally trapped disturbances envisages these disturbances evolving into a propagating coastal gravity current with associated nonlinear coastal waves from an initial marine layer disturbance. This disturbance, which is an area of raised marine layer for the North American and Australian coastal ridges, is generated by the interaction of some synoptic scale feature with the coastal mountains (in this case the ridging of a Southern Ocean anticyclone along the south coast of Australia). If significant propagation as a gravity current is to occur, then this initial disturbance must be large enough to act as a reservoir of dense fluid (see Chapters 2 and 3). First, the observed propagation speeds of the ridge (40 m/s on the south coast and 20 m/s on the east coast, Holland and Leslie (1986)) are compared with the theoretical Kelvin wave (2.57) and gravity current speeds (2.11). To apply (2.11), the observed pressure difference between the gravity current air and the ambient 153 atmosphere is required. From Holland and Leslie (1986), this difference is 4-5 mb so that (2.11) gives the computed gravity current speed as at most 16.1 m/s, using an ambient air density of 1.2 kgm . It is clear that this gravity current speed is far smaller than the observed speed on the south coast of 40 m/s. Similarly, the linear Kelvin wave speed (g'H)-'-/^  = 30 m/s, computed from g' = 0.6 ms"^  and H = 1500 m (Holland and Leslie, 1986), is also seen to be too small. Comparison of the observed speed with the solitary Kelvin wave speed (2.57) requires an estimate of a, the ratio of the amplitude of this wave to the total depth of the fluid layer. From Figs. 5.4 a-c, Holland and Leslie (1986) state that, following the arrival of the ridge, the top of the stable layer lifted at Mascot station (at the southern end of the east coast - see Fig. 5.1) from 850 mb to 700 mb. These authors state that radiosonde data shows this magnitude of increase in the lower layer depth also occurs at Wilson's Promontory on the south coast but do not provide the actual data. Applying equation (3.1) gives the thickness of the raised layer as 1704 m and hence the amplitude ratio a = 0.57. Performing exactly the same calculation for Nowra station, just to the south of Mascot, gives a = 0.49 which is somewhat less than the Mascot figure. As discussed later, this discrepancy in the observed amplitude ratios may be due to the dissipative effects of gaps in the coastal mountains (see Fig. 5.1). Substitution of a = 0.57 (0.49) into equation (3.2) shows that the theoretical solitary Kelvin wave speed is 37.6 (36.6) m/s. This figure agrees well with the observed speed on the south coast of 40 m/s. On the east coast, Holland and Leslie (1986) report that g' = 0.3 ms"^  and H = 1200 m so that the theoretical linear Kelvin wave speed is 19 m/s. This value agrees well with the observed speed of 20 m/s. Applying (2.57) indicates that the theoretical solitary Kelvin wave speed is 23.8 (23.2) m/s at Mascot (Nowra). Further north at 154 Williamtown, which from Fig. 5.1 is situated near a large gap in the coastal mountains, and Brisbane, the amplitude ratios are calculated from (3.1) as 0.52 and 0.18 so that the theoretical solitary Kelvin wave speeds are 23.4 m/s and 20.6 m/s respectively. These values are seen to be in reasonable agreement with the observed speed of 20 m/s. Maximum probable errors are estimated from the uncertainties in Holland and Leslie's (1986) data as ±2.2 m/s for the gravity current and linear Kelvin wave speeds and ±2.8 m/s for the solitary Kelvin wave speed. These speeds are summarised in Table 5.1. Note that the above calculations have indicated that both the linear and solitary Kelvin wave speeds compare reasonably with the observed speed of the ridge on the east coast, but that only the latter agrees well on the south coast. From Chapter 2, it is expected that the ridge should take a solitary rather than a linear Kelvin wave form. This expectation is supported by several of the observations. For example, the ridge is seen to be of large amplitude (Fig. 5.6 and the ratios calculated above) which means that nonlinearities are important. Also, Holland and Leslie (1986), the ridge is strongly dispersive in that as it propagates northward, the ridge loses its structure. For example, these authors state that there is a loss of sharpness in the wave front and an amplitude decrease as the ridge propagates. While the former example is not disputed, it is felt that the observed amplitude decrease occurs mainly through the dissipative (see below) action of mountain gaps and friction rather than via dispersion. Hence, with both nonlinear and dispersive effects being observed, it is clear that the solitary Kelvin wave model which requires both these effects, rather than a linear Kelvin wave which contains neither, is consistent with the observations. In the following analysis, the coastal ridge is therefore treated as having a solitary Kelvin component. 155 T a b l e 5.1 E v o l u t i o n t imes and speeds of p r o p a g a t i o n of the A u s t r a l i a n c o a s t a l r i d g e . Observed Computed Maximum E r r o r Range Evolution time (hrs) 24 50 42-58 Speed, south 40 coast (m/s) G-C 16.1 LKW 30.0 SKW 37.6 13.9-18.3 27.8-32.2 34.8-40. 4 Speed, east coast (m/s) 20 G-C 16.1 LKW 19.0 SKW ( Wf 23 . 4 SKW (B) 20.6 13.9-18.3 16.8-21.2 20.6-26 . 2 17.8-23.4 Notes . G-C LKW SKW (W) (B) gravity current linear Kelvin wave s o l i t a r y Kelvin wave Wi 11iamtown Br isbane 156 The theoretical evolution time (3.3) of this solitary Kelvin wave is now compared with observations. Using g' = 0.6 ms"2 and H = 1500 m for the south coast (Holland and Leslie, 1986), one calculates from (2.8) that R = 327 km. The alongshore length . scale is estimated from the product of the observed propagation speed of 20 m/s (Holland and Leslie, 1986) and the time span of propagation. The latter is taken as 16 hours from the surface pressure traces in Fig. 5.6 so that L = 1152 km. Finally, from the amplitude ratios calculated above, the ratio H/a is estimated as 2. Substitution of these parameters into (3.3) gives an evolution time of 50 hours (maximum possible error ±8 hours) for the solitary Kelvin wave. This value is seen to be somwhat larger than the maximum of 24 hours suggested by the surface pressure traces (Fig. 5.6), the soundings (Fig. 5.4) or the satellite imagery (Fig. 5.2). From above and the analysis of Chapter 2, it appears that the first question posed by Holland and Leslie (1986) should not be one of which component (gravity current or Kelvin wave) dominates but rather one of what is the interaction between these components, and what contribution is made by each to the overall dynamics. This analysis has indicated that for substantial propagation of the ridge as a gravity current, a large reservoir of raised marine layer must form initially. Satellite imagery (Fig. 5.2a) shows that the initial disturbance in the eastern Great Australian Bight is confined to a narrow coastal zone probably less than a Rossby radius (327 km) in width (compare with width on east coast in Fig. 5.2b). Also, the initial disturbance appears somewhat narrower in offshore extent than that of the July, 1982 Californian gravity current event (Section 3.4, Appendix 2, Figs. 3.15a-b). Thus, qualitatively, it seems that the initial disturbance in the Bight is too small to allow significant propagation as a gravity current. 157 Consistent with this view is the shorter evolution time (24 hours) observed than that for the July, 1982 and May, 1985 gravity current events (48 hours, Appendix 2). The possibility that the nonlinear coastal wave might steepen and form a shock instead of a solitary Kelvin wave is not considered likely here because the observations do not show the significant across-shore winds at the leading edge of the ridge that are needed for a shock wave (Kubokawa and Hanawa, 1984; Chapter 3). For example, at Williamtown, Fig. 5.5 shows that the winds switch to southerly or alongshore as the ridge arrives while just ahead of the leading edge the winds are light and variable and represent a local sea breeze. This situation is observed all along the east coast whereas on the south coast the winds are entirely alongshore (Holland and Leslie, 1986). Other data and a numerical simulation using a fine mesh, primitive equation model (Holland and Leslie, 1986) support this view of the ridge as having a solitary Kelvin rather than a shock wave component. Thus, the above argument implies that the coastal ridge should be regarded as having been formed initially as a coastal gravity current which then evolved a solitary Kelvin wave component. As shown, both theoretically and experimentally by Kubokawa and Hanawa (1984) and experimentally by Maxworthy (1983) and Simpson (1987), this solitary Kelvin wave propagates more quickly than the gravity current part so that it is expected to move through and ahead of the latter. This viewpoint is entirely consistent with Holland and Leslie (1986), who in their fine mesh primitive equation model, found evidence of a disturbance moving through and ahead of the main body of the coastal ridge. It is also consistent with the observed speed of the ridge being much faster than the theoretical gravity current speed but similar to the solitary Kelvin wave speed. On the other hand, the displacement of the marine layer behind the leading edge, as detected from Figs. 5.4a-c, 5.5, 5.6, appears more indicative of a gravity current than a Kelvin wave because the profiles seem to 158 display the steep leading edge and bulbous head typical of gravity currents (Simpson, 1987) rather than a wavelike displacement. To address the second question of Holland and Leslie (1986) concerning the fact that the ridge could propagate around the convex bend in the coastal mountains near Gabo Island but not at Brisbane, equation (3.5) is applied to show that this behaviour is consistent with a solitary Kelvin wave. Application of (3.5) for a = 0.57 (0.49) observed at Mascot (Nowra) and inferred for Wilson's Promontory give a critical angle of 75° (70°). From Fig. 5.6, the angle of the convex bend in the mountains (1000 m contour) at Gabo Island is seen to be about 65° so that theoretically, a solitary Kelvin wave of this amplitude should be able to propagate around without loss of identity. On the other hand, at Brisbane, a = 0.18 so that the critical angle from (3.5) is 42°. Fig. 5.6 indicates that from Brisbane onwards, there are no mountains of 1000 m height until the range just south of Townsville (about 700 km to the north of Brisbane) and that the 600 m contour bends about 50°. Thus, it is expected that a solitary Kelvin wave of this amplitude will not be able to propagate further than Brisbane but instead, will separate from the coastal mountains and lose its identity. Fig. 5.2b indicates that this is precisely what happens. Instead of propagating further, the overlying cloud deck associated with the coastal ridge is observed to diffract at Brisbane and decay out to sea. Further confirmation of the inability of the ridge to propagate beyond this point can be seen from Fig. 5.6, since Rockhampton, some 400 km north of Brisbane, shows no record of the pressure rises observed to the south. In summary, it is seen that the observed behaviour of propagation of the ridge around the convex bend at Gabo Island but not at Brisbane is consistent with a solitary Kelvin wave. The effects of other, smaller coastal irregularities (question three above) are now considered. 159 From Fig. 5.6, it is evident that there are other substantial bends (both concave and convex) as well as gaps (river valleys) in the coastal mountains of southeastern Australia. According to Holland and Leslie (1986), these topographic irregularities are largely responsible for the decrease in amplitude of the coastal ridge and loss in sharpness of the wavefront as this disturbance propagated northwards. In particular, the most pronounced weakening (i.e., reduced displacement of the lower layer and less pronounced wind shifts and southerly winds, see Holland and Leslie, 1986) of the coastal ridge was observed to occur in the vicinity of large valleys. The effect of small (radius of curvature much greater than the Rossby radius) coastline irregularities on a propagating linear, unforced Kelvin wave has been considered by several workers, e.g., Miles (1972); Mysak and Tang (1974). Clarke (1977a, 1977b) extended this analysis to include both forced and nonlinear Kelvin waves. In general, the observed weakening of the coastal ridge near valleys is consistent with the analyses of Mysak and Tang (1974) and Clarke (1977a, 1977b). The latter also showed that at a convex bend or "cape", a nonlinear, but not solitary, forced Kelvin wave should slow down and show an increased displacement while at a concave bend or "bay" this wave should exhibit the reverse. As far as a solitary Kelvin wave is concerned, the effect of small (i.e., less than the critical angle given by equation (3.5)) coastline irregularities may be considered as follows. For a coastline that is slowly varying in the sense that the radius of curvature of the variations is much greater than the typical length scale of the solitary wave, the problem becomes one of this wave propagating in an inhomogeneous medium which leads to the evolution equation no longer being a simple KdV equation, but rather a KdV with variable coefficients (Grimshaw, 1983). The leading order solution of this variable KdV equation was shown by this author to be the familiar asech^kT solution of the standard KdV equation, where now the amplitude a is not constant, as for the 160 standard KdV, but a function of the slow time variable. Thus, both the amplitude and hence the local phase speed of the solitary Kelvin wave may now vary slowly as the topography is modified on this small scale (Grimshaw, 1983). Further analysis (Grimshaw, 1983) when a viscous dissipation is included (so that the variable KdV becomes a variable KdVB equation), indicates that the amplitude is expected to decay as 1/T, where T is the slow time scale given by (3.3). A KdVB equation (equation 2.67 and A.12) can also be derived (Appendix 1) for the modified Kubokawa and Hanawa model for the case where there is a porous boundary, i.e., where there is a constant loss of energy from the solitary Kelvin wave component due to gaps in the coastal mountains. Fig. 5.6 indicates that there are indeed several large gaps in the southeastern Australian mountains, particularly near Williamtown and Nowra. It is interesting to note that at both these stations, the solitary Kelvin wave amplitude is less (0.49 and 0.52) than its computed value at Mascot (0.57) which lies almost exactly halfway (about 100 km) between these two stations. Unfortunately, there is no way to determine from the sources available whether this decrease is due to the gaps in the mountains or to errors in the data. Certainly, the decrease of amplitude from Williamtown (a = 0.52) to Brisbane (a = 0.18) is too large to be a data error. Consequently, it is felt here that the observed amplitude decreases probably arise through the dissipative effects of gaps in the mountains and to a lesser extent viscosity. In both cases, the evolution equation for the solitary Kelvin wave component becomes a KdVB equation. As mentioned previously, a KdVB evolution equation leads to an amplitude decay of l / T , where T is the long time scale given by (3.3). It is shown now that this predicted decay is consistent with observations. First, the expected time that was taken by the ridge to propagate from Nowra station to Brisbane is computed. From 161 the observations, Holland and Leslie (1986) calculated the average propagation speed of the ridge on the east coast to be 20 m/s. This value then implies a travel time of 14 hours for the approximately 1000 km separating Brisbane from Nowra. Hence, for a 1/T decay (where from above T = 50 hours), the predicted amplitude at Brisbane is a factor of 50/(50 -f 14) of the observed value at Nowra of 0.49. Thus, the predicted Brisbane amplitude is 0.38 (or 0.31 if the observed T = 24 hours is used) as compared to the observed value of 0.18. Despite the discrepancy between the observed and computed values, which probably arises because the theory makes the gross assumption of a constant dissipative loss all along the propagation path, the KdVB model is felt to be sufficiently robust to indicate the probable importance of gaps in the mountains on the coastal ridge. Further confirmation of the appropriateness of the KdVB model comes from a comparison of the displacement of the pressure and temperature fields (Figs. 5.4a-c and 5.6) with the displacement structure expected from the model. For long enough times, Johnson (1970, 1972), Whitham (1974) and Smyth (1988) showed that the KdVB equation has a steady, undular bore solution. Fig. 5.7, taken from Whitham, illustrates one such steady state solution for the case where the dissipation allows an oscillatory (i.e., undular) decay to the asymptotic end state rather than the monotonic decrease for stronger dissipation. Unfortunately, the available data do not allow all the coefficients in the KdVB equation to be evaluated so it is not possible to determine the strength of the dissipation relative to the nonlinear and dispersive terms in the model. Qualitatively, however, the remarkable similarity between the undular bore solution of the KdVB equation (Fig. 5.7) and the observations (Figs. 5.4a-c and 5.6) provides further evidence of the appropriateness of this model for the propagation of the coastal ridge along the east coast between Nowra and Brisbane. 162 Figure 5.7. Steady state, undular bore solution of the Korteweg-deVries-Burgers equation. The normalised variables z and i are proportional to interface displacement and wave front propagation distance respectively. Adapted from Whitham (1974). 163 Finally, a few brief comments are made concerning the fourth question raised by Holland and Leslie (1986) about the possible effects of coastal temperature gradients. It would seem that the most important effect of these would be the generation of local sea breezes which are cited by these authors as maintaining the lower stable layer in which the coastal ridge propagates. Fig. 5.5, discussed above, has shown that ahead of the ridge a light and variable circulation, indicative of local sea breezes, occurs but that once the ridge has passed the winds switch to southerly in agreement with Kelvin wave theory. On a larger scale, coastal temperature gradients (both along- and across-shore) may cause inhomogeneities in the waveguide, in the form of changes in the stratification, which may then lead to local variations in the propagation speed of the coastal ridge. In more tropical areas, for example near Brisbane, greater surface heating may contribute to the cessation of propagation by weakening the stratification and hence the trapping of the coastal ridge. This situation is analogous to that discussed for the Southern African coastal low in Chapter 4 in which the likelihood of less efficient trapping and slower propagation of the disturbance on the warm Agulhas Current coast was considered. 5.4 Conclusion Observations of a representative case study (9-11 November, 1982) of a coastal ridging event in southeastern Australia, given in Holland and Leslie (1986), have been analysed further in light of the modified Kubokawa and Hanawa model of Chapter 2. It is considered that the coastal ridge evolved initially as a gravity current from a marine layer disturbance in the Great Australian Bight. This gravity current then developed a solitary Kelvin wave component which was expected to propagate 164 through and ahead of the gravity current. Such an expectation is consistent with the observations of Holland and Leslie (1986) and with a fine mesh, primitive equation numerical simulation performed by these authors. It is also consistent with observations obtained from rotating tank experiments (Maxworthy, 1983; Kubokawa and Hanawa, 1984; Simpson, 1987). Good agreement was found between the observed propagation speed of the ridge and the solitary Kelvin wave speed. Other observed features of the coastal ridge that were explained by the model include the ability of this disturbance to travel around the convex bend in the coastal mountains at Gabo Island without separation and the cessation of propagation at Brisbane. Also, the model could account to some extent for the observed decrease in amplitude of the coastal ridge as it propagated northwards. It was argued that this decrease was mainly due to gaps in the coastal mountains. In conclusion, it may be said that the work presented in this chapter has largely resolved three of the four questions posed by Holland and Leslie (1986). Thus, it is felt that this chapter has clarified the relationship between the gravity current and solitary Kelvin wave components of the coastal ridge, has provided a theoretical explanation for the observed propagation of this ridge around the convex bend in the coastal mountains at Gabo Island but not at Brisbane, and has shown qualitatively (and to a first approximation, quantitatively using a KdVB model) how smaller irregularities (e.g., gaps) in these mountains act to dissipate the coastal ridge. The fourth question, which has only been commented on here, concerns the effects of coastal temperature gradients. It was noted that these were important for generating sea breezes and for locally modifying the propagation speed. The question of how important these sea breezes (as compared to large scale anticyclonic subsidence) are 165 for maintaining the waveguide for coastal ridge propagation cannot even be partially-answered with the data available. It is considered here that the attempted resolution of this question represents a high priority for future research. 166 Chapter 6. Summary and Conclusions In this thesis, the dynamics of coastally trapped disturbances that propagate in the marine layers of western North America, Southern Africa and southeastern Australia have been examined. It has been shown that, despite some obvious differences in the nature and form of propagation of these disturbances in the three different regions, the underlying fundamental dynamics is the same in each case. It was argued that the observed differences between coastally trapped disturbances in the three regions was rather a function of the different types of forcing and boundary conditions present in a given region than anything intrinsic to the marine atmosphere itself. For example, in each case the disturbances were shown to be trapped in the marine layer from above by the presence of a strong subsidence inversion on top of this layer. In the horizontal, the coastal mountains of each region acted as a barrier against any penetration of the disturbance inland and trapping of the disturbance offshore to within a Rossby radius of these mountains was achieved through Coriolis effects on the alongshore flow. This horizontal trapping, together with the observation that the disturbances always propagated with the coastal mountains on their right (left), looking downstream, in the Northern (Southern) Hemisphere was indicative of Kelvin dynamics. Forcing for the disturbances was provided through the interaction of some synoptic feature with the coastal mountains. It was argued that the location of the three regions studied was particularly favourable.because it was subtropical and hence subjected to large scale subsidence through the presence, especially during the summer, of semi-permanent anticyclones over the adjacent oceans. Also, the subtropical location was still far enough from the equator that the regions could be 167 influenced by synoptic perturbations in the mid-latitude westerlies which would provide forcing for the disturbances. The other favourable feature of these regions was that they all had substantial coastal mountains that were oriented roughly orthogonal (i.e., north south) to the zonal movement of these perturbations in the westerlies. Based on these arguments, it can be inferred that other favourable locations on the Earth where these disturbances might be found are the subtropical coastal areas of Chile and Northwest Africa. Finally, it was argued that coastally trapped disturbances are mesoscale phenomena by virtue of their length scales being in the range 100 - 2000 km both offshore (a Rossby radius) and alongshore and by both rotation and nonlinearities being important in their dynamics. On the basis of this commonality between coastally trapped disturbances in the three regions, an attempt at a general theory of these disturbances was made. The governing equations for this theory were those for a shallow water, flat-bottomed model on an f plane, with a reduced gravity representation for the density stratification of the lower marine atmosphere. The other fundamental approximation that was made was to use the semigeostrophic version of these equations valid for motions that have their alongshore flow in geostrophic balance with the across-shore pressure field but whose alongshore pressure field is not geostrophically balanced. This approximation was justified by virtue of the observed across-shore scale (a Rossby radius) and velocity being much smaller than the alongshore ones (100 - 300 km compared to 1000 - 3000 km and 1 m/s compared to 10 m/s). Application of the approximation was affected through a perturbation expansion that was based on a small parameter defined in terms of the square of the ratios of the Rossby radius to the alongshore length scale and a wave amplitude to the lower layer depth. It was shown that, in the simplest linear case, both a Kelvin wave and a coastal gravity current were acceptable solutions to the equations but that when higher order effects 168 were included an intruding coastal gravity current evolved nonlinear coastal (i.e., Kelvin) and frontal waves which, for long enough times, could develop into solitary Kelvin and frontal waves respectively. To determine whether nonlinearities were important in the dynamics, the Rossby and Froude numbers of both the incident forcing flow and the resulting motion of the coastally trapped disturbance was computed for each case. It was found that the South African coastal low differed considerably from the North American and Australian disturbances because it was characterised by weak blocking of the incident flow and by weak nonlinearities. On the other hand, the North American and Australian coastal ridges were typified by stronger blocking and nonlinearities. Another important difference between the disturbances was that the South African coastal low was coritinously forced throughout its propagation path and did not develop, like the North American and Australian coastal ridges, from a large area or reservoir of raised marine layer directly forced by the interaction of a synoptic system with the coastal mountains. In the former case, the continuous forcing consisted of a buoyant influx of warm air driven off the interior escarpment of Southern Africa in a seawards direction by the interaction of the synoptic forcing with this escarpment. Vorticity arguments then showed that a coastal low rather than a coastal high should form. For the North American and Australian cases, the approach of a synoptic system towards the coastal mountains in a bight (Southern Californian or Great Australian Bights) directly lifted the marine layer there to form a reservoir for subsequent development of a propagating coastally trapped disturbance. The size of this reservoir then helped detennine whether the disturbance propagated initially as a gravity current or as a Kelvin wave. In any case, it was considered that the disturbance evolved eventually into either a solitary Kelvin wave or a shock Kelvin 169 wave depending on whether dispersive effects were important. Unlike the South African coastal low, the initial flow was either on - or alongshore and hence the disturbance formed as a mesoscale ridge of high pressure rather than as a trough. Thus, it was argued that the Australian coastal ridge quickly evolved a solitary Kelvin wave component quickly from its gravity current part. For the North American coastal ridges, the July, 1982 and May, 1985 events propagated as gravity currents with associated shock Kelvin waves while the May, 1982 event propagated as a solitary Kelvin wave right from its initial stages due to the localised and impulsive nature of the forcing in this event. In all these examples, the interaction between a mean gravity current and an evolving nonlinear Kelvin wave were important. For the South African coastal low however, this was seen not to be the case and the disturbance was treated primarily as a forced, linear Kelvin wave with only very weak nonlinear influences.-Based on the above views of the dynamics of the coastally trapped disturbance in each region, theoretical predictions of the scales of the initial marine layer reservoir, and characteristics of the disturbance such as the evolution time scale, the propagation speed and the width, and the perturbations of the marine layer winds, temperatures and pressure fields associated with its propagation were all compared with data given in existing studies of these phenomena and, in the case of South Africa and California, previously unanalysed data. In each case, good agreement was found between the appropriate theoretical model and the observations. In addition to their influence in determining the nature of the propagation, the forcing and boundary conditions of each region were also seen to be important in controlling, to large extent, the local manifestation of the disturbance and ultimately, 170 the lifespan. For example, in the case of South Africa, the effects on the south and east coasts of greater surface heating due to the warm Agulhas Current and advection by the trailing mid-latitude frontal system were likely responsible for the observed greater speeds of the coastal low and higher inversion levels here than on the cool water west coast. It was also suggested that bends in the coastal mountains in the southwest of this region could lead to local modifications of the phase speed of a coastal low propagating as a linear Kelvin wave whereas in the northeast, downstream of Durban, the convex bend inland of the mountains and the weakening of the subsidence inversion in these more tropical areas discouraged further propagation of the coastal low. In some cases however, the coastal low was dissipated by the trailing frontal system before it could reach Durban and it was suggested that this was more typical of winter when these fronts track nearer the landmass. For the Australian coastal ridge, the most prominent regional effects were considered to be the convex bend in the coastal mountains near Brisbane which caused cessation of the disturbance there and to the south, gaps in the mountains near Sydney. The latter were thought to exert a dissipative influence and thus be responsible for the observed decrease in amplitude of the coastal ridge as it propagated northwards towards Brisbane. This dissipative influence on the coastal ridge was modelled directly by treating the coastal mountains as a porous rather than as an impermeable barrier. Resulting from this treatment, a KdVB equation rather than a KdV equation was derived for the solitary Kelvin wave component of the ridge and reasonable agreement was obtained between the theoretical and observed amplitude decrease. In western North America, the convex bend in the coastal mountains at Cape Mendocino was found to play a crucial role in causing the cessation of propagation of the May, 1982 solitary Kelvin wave disturbance. This cessation occurred because the 171 critical angle, which depends directly on the solitary Kelvin wave amplitude, through which this wave can turn without separation from the coastline was seen to be smaller than the actual bend in the coastal mountains there. For the gravity current cases of July, 1982 and May, 1985 cessation of propagation likely occurred through a combination of factors since there was no known fundamental reason in the dynamics why the disturbance should not have continued propagation. Both cases were able to propagate well past Cape Mendocino to reach Cape Blanco and the northwestern tip of Vancouver Island respectively. It was argued that it was the combination of the irregularity in the topography and the strength of the opposing mean winds at these locations together with the weakening of the forcing reservoir in the Southern Californian Bight at about the time that the disturbance reached their end-points that prevented further propagation. From the foregoing, it may be said that the attempted general theory of coastally trapped disturbances developed in this thesis has been able to account, to some extent, for most of the observed salient features of these phenomena. The theory has also provided some foundation for the largely descriptive treatments by Dorman (1985, 1987, 1988) and Mass and Albright (1987, 1988) of these disturbances in western North America and largely resolved the dispute between these authors over the interpretation of the May, 1982 event as a solitary Kelvin wave and a coastal gravity current respectively. In the case of the Australian coastal ridge, this theory is considered here to account for the observation in both the data and the numerical simulations of Holland and Leslie (1986) of a solitary Kelvin wave component propagating through and ahead of the main gravity current feature. For the South African CcLSC, ei linear, forced Kelvin wave model was developed to show that the disturbance evolved as a coastal low rather than as a coastal high and it was 172 considered that the model extended the theoretical treatments of this disturbance by Gill (1977), Anh and Gill (1981) and Bannon (1981). Having summarised the results of the thesis, some suggestions for future research are now made. On the observational side, there are several areas in which future efforts could be concentrated. Firstly, from the point of view of obtaining a global understanding of coastally trapped disturbances, an ideal objective would be to obtain surface weather observations and satellite images for the other areas of the world considered particularly favourable for the propagation of these disturbances. Should these data indicate the likely existence of these disturbances, then an effort to obtain higher resolution data (e.g., soundings) could be made. As argued above, the subtropical coastal areas of Chile and northwestern Africa would seem the best candidates. Indeed, as mentioned in Holland and Leslie (1986) and Mass and Albright (1987), synoptic pressure maps for South America do indicate the likely existence of coastally trapped features that originate in Chile and propagate around the southern tip of the Andes into Argentina. Secondly ^  the existing studies of coastally trapped disturbances in western North America, Southern Africa and southeastern Australia could be improved through the acquisition of better resolution data which would be able to provide more details of the finer structure, particularly near the leading edge, of the disturbance. Thus, it would be preferable to supplement the existing weather network with data obtained through judiciously placed acoustic sounders and aircraft flights. The possibilities of obtaining ship of opportunity data or collaborating with coastal oceanographic experiments to obtain data for the offshore marine layer would be very worthwhile from the viewpoint of resolving the trapping width and potential frontal nature of the disturbances which are, at present, detected mainly by satellite imagery. 173 A better understanding of the disturbances may also be achieved through well-designed rotating tank experiments similar to those conducted by Griffiths and Hopfinger (1983), Maxworthy (1983) and Kubokawa and Hanawa (1984). These authors have been able to show the evolution of a solitary Kelvin wave from an intruding coastal gravity current in the laboratory and it may be possible to extend their work further to simulate the effects of topographic irregularities like Cape Mendocino and the formation of the gravity current by the interaction of some large scale flow with the topography (i.e., a ridge parallel to the side wall in the rotating tank context). On the theoretical side, future efforts should be devoted towards a better understanding of the effects of some of the forcing and boundary conditions. For example, a more detailed model of the generation of the coastally trapped disturbances by the interaction of the forcing synoptic systems with the coastal mountains needs to be developed. In the North American case, the relationship between the scale of the synoptic forcing and the potential formation of a reservoir in the Southern Californian Bight and hence propagation of the disturbance as a solitary Kelvin wave or as a gravity current should be investigated further. For the Australian coastal ridge, the blocking of the forcing anticyclone by the coastal mountains in western Victoria, considered by Baines (1980) in his model of the Southerly Buster and hinted at by Holland and Leslie (1986), plays a role in the generation of the coastal ridge which is not well understood at present. Other theoretical improvements to the understanding of coastally trapped disturbances include the consideration, in more detail than herein, of the effects of dissipation, both through gaps in the coastal mountains and surface friction, and 174 those of land-sea temperature contrasts. Thus, the effects of local modification in the waveguide of the disturbances by topographic irregularities should be considered further. In conclusion, it may be said that the above research suggestions are not the only ones nor even the most obvious ones. Since the study of the dynamics of coastally trapped phenomena in the atmosphere has only been attempted fairly recently (Gill (1977) being the first paper) and does not possess an extensive literature, there is much that can be done to improve the collective knowledge of these features. On the other hand, in the oceans, coastally trapped phenomena have been studied extensively and for some time. It is felt here that, since both media are rotating stratified fluids, there are many analogies to be drawn and much to be learned in coastal meteorology from a familiarity with how these phenomena behave in the ocean. Thus, a better diagnostic understanding of coastally trapped disturbances in the atmosphere may be achieved through guidelines laid down by the study of these phenomena in the oceans. Once this understanding has been achieved, the ideal objective of accurate predictions of the behaviour of the disturbances through a fine resolution numerical model and ultimately, the routine forecasting of the coastal weather caused by these disturbances may be accomplished. 175 References Anh,N.N., and A.E.Gill, 1981: Generation of coastal lows by synoptic-scale waves. Q. J. R. Meteorol. Soc, 107, 521-530. Baines,P.G., 1979: Observation of stratified flow over two-dimensional obstacles in fluid of finite depth. Tellus, 31, 351-371. - , 1980: The dynamics of the southerly buster. Aus. Meteorol. Mag., 28, 175-200. Bannon,P.R., 1981: Synoptic-scale forcing of coastal lows. Forced double Kelvin waves.in the atmosphere. Q. J. R. Meteorol. Soc, 107, 313—327. 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Wiley (Interscience) New York. 181 Appendix 1: Wave evolution equation for the dissipative case The purpose of this section is to derive an evolution equation for the wave amplitude that describes the dissipation of the energy of coastally trapped disturbances due to either leaks or gaps in the coastal mountains or viscous losses in the marine layer as a whole. From (2.43-2.48) and following Kubokawa and Hanawa (1984), consider solutions consisting of a semigeostrophic component and a small ageostrophic component, i.e., let (Al) (A2) (A3) where v g o , h g o are the components of vector V Q , vs^, hg} are components of vector V-j_ and v a, h a are the components of vector V^. The vectors V Q , Vj and V 2 are related to V, defined in the characteristic equation (2.48), by v = v 0 + d 2y 1 + d 4v 2 + .... Then (2.38-2.40) become ( vsl)t + vso( vsl)y + ((vso)x •+ i K l + (hsl)y ( A 4) •+ d 2 { v s l ( v s l ) y + u sl( v sl)x + ((va)t + vso(va)y + ((vS G)x + l)u a + (ha)y} = 0(d4) .( usl)t + vso( usl)y + v a + (ha)x = ° ( d 2 ) ( A 5) K l ) y - ( v a ) x - k 2 h a = 0(d2) (A6) Note that the forcing synoptic pressure gradient has been neglected for simplicity here. From the method of characteristics, 9 ^  = -c_j_ 9 x + 0(d2), where c_j- are the eigenvalues of the matrix A in equation (2.48). Substituting in (A5) and (A6) gives v a + (ha)x = " v s o(y)}V(x)r y y + 0(d2) (A7) (v a) x + k 2 h a = V(x)r y y + 0(d2) (A8) where V(x) = {(vso(x) - c ± ) R . V V o v s o + R .Vv 0h S 0}/k 2h S 0(x) (A9) V y 0 = (9 / 9 V 0 , 9/ 9LQ) and V Q , L Q are the components of vector V defined in equation (2.48) R = right eigenvector of matrix A = RT.+ 0(d2) 183 Hence (Kubokawa and Hanawa, 1984), v a and h a can be written as v a = f(x)r y y + 0(d2) (A10) K = s(*)r y y + o(d2) From (A6) and (A8) and the conservation of potential vorticity (2.42), u sly = V ( x ) r yy (vso)x + 1 = " k 2 h so so that u((vso)x + X) =- k 2 h so u = 0 at x = - L Q (density front) = - d 2 k 2 h s o u a at x = 0 (leaky mountains) Equation (A4) then becomes at x = 0, (va)t + vso(va)y + (ha)y " d 2 k 2 h s o u a = -d"2((vsl)t ( A 1 1 ) + v s o ( v s l ) y + ( n sl)y> " v s l ( v s l ) y - u s l ( v s l ) x +. ° ( d 2 ) 184 Substituting for the solutions for V i , v g i , h si obtained by Kubokawa and Hanawa (1984) using standard reductive perturbation techniques, namely, Vj_ = Rr + 0.5d2(R.V y0)RT2 + °(d4) v s l = ^ V Vovso + O . S d ^ . V V o ) 2 V s o + 0 (d 4 ) • hi = Y i - V V o h S o + ^2(Xl^Vo)\o + 0( d 4 ) and (A10) into (All) then gives at x = 0 (i.e., at the leaky mountain barrier) r T + a i r T + pr T T T = k 2 ^r T T + o(d4) . (A12) where a = R.V V o c ± - V o p, = d 2k _ 2{l - cosh kL Q + kVGsinh kLQ},.> 0 (3 = constant, given by equation (4.9) of Kubokawa and Hanawa (1984). T = y - c ± t Equation (A12) is a Korteweg de Vries Burgers (KdVB) equation which is seen to be satisfied by the nonlinear Kelvin wave at the coastal mountains (x = 0). At the density front, x = - L Q , the above substitutions into (All) give the KdV equation derived by Kubokawa and Hanawa (1984), i.e., equation (A12) with \i = 0. Equation (A12) is also derivable for the case of solitary wave subject to viscous losses in the fluid medium as a whole (see Grimshaw, 1983). 185 Appendix 2: Other North American Coastally Trapped Events. 1. Californian Event of 13-20 July, 1982 Like the May, 1982 case discussed above, the July, 1982 event was generated in the Southern Californian Bight following the passage of a synoptic scale feature westwards across the coastal mountains. Dorman (1987) considers this case to be representative of the unsteadily progressive events that propagate in the Californian marine layer aperiodically during the summer and which are best described as coastally trapped gravity currents. The generating synoptic feature in question here was a large baroclinic trough at 700 mb of about 1000 km in width and some 2000 km in meridional extent (see Fig. Al). Forcing on this scale was able to generate a large area of raised marine layer (see Fig. 3.15, 3.16) in the Southern Californian Bight which could then act as a substantial reservoir for coastal gravity current propagation. Propagation of the generated marine layer disturbance then occurred from Point Conception (just south of Vandenburg AFB) well past Cape Mendocino to as far as Cape Blanco (see Fig. 3.2a for locations of place names) in southern Oregon. Soundings taken in the Bight area at San Diego (Fig. A2a) and Point Mugu (Fig. A2b) show the inversion lifting to about 750 - 800 m on July 16th following the approach of the trough towards the coast. Figs. Al-2 also indicate that onshore winds occurred in the Bight which is consistent with the event forming as a coastal ridge rather than as a trough (see Section 2.4). Details of the synoptic situation and the surface weather changes associated with the propagation of the disturbance are available in Dorman (1987). Suffice it to say here 186 Figure A l . The geopotential height (metres) of the 700 mb surface in the atmosphere on July 16, 1982 at 1200 UTC. Note the trough, which forced the July, 1982 event, crossing the coast. Adapted from Dorman (1987). 187 U/7(0). 12/7(12) -20 -to 0 10 20 TEUP (DCC C) '3/7(12) -20 - io 0 to 20 TEUP (DCC C) '3/7(0) -10 0 10 20 TEUP (DEC C) -20 -10 0 10 TEUP (DEC C) 15/7(0) "S V _ 4000 • / \ an 1 3 5 0 0 7 \ , 3O0O \ \ 3 2500 \ \ < i 2000 \ \ r - 1500 \ § 1000 2 I 300 i < 0 13/7(12) -10 0 to 20 T E U P (DEC C) 1 « / 7 ( I 2 ) '6/7(0) -20 -10 0 10 TEUP (DEC C) 17/7(0) -10 0 10 TEUP (DEC C) -30 -20 - ' 0 0 10 20v TEMP 'DEC C) -10 0 to TEUP (DEC C) Figure A2a. Temperature (right hand curve) and dewpoint temperature (left hand curve) soundings for San Diego during the period 00 UTC July 12 to 00 UTC July 23, 1982. The base of the inversion is indicated by an arrow on each sounding. 1 8 8 l » / 7 { 0 ) 18/7(12) 19/7(0) 0 : 10 TCUP (DCC C) 19/7(12) -30 - 20 -10 0 10 TEUP (OCG C) 20/7(0) -20 -<C 0 10 20 TEU5 (DCC C) 4300 -20 -10 0 10 20 TEMP (DEC C) 20/7(12) -30 -20 -10 0 10 20 30 TCUP f«c C) 4000 [ 3S00 ' 3000 [ 2500 !• 2000 - 1500 > 1000 500 0 1 7 \ < / ( \ ' \ -10 0 10 20 TEUP (DEC C) 21/7(0) 21/7(12) 2i/7(0) 10 20 UUP (DCC C) 22/7(12) 0 10 20 U U " (DEC C) S 10 15 20 2< TCUP (DEC C) 0 10 20 TEUP (DEC C) Figure A2a Continued 189 4500 4000 m 1 , 3000 3 2300 o Z, 2000 500 0 13/7 (ti) \ < . \ \ < • \ \ < \ \ * V Y \ —/*— -20 - 1 0 0 10 TEUP (DEC O 20 4300 4000 3> | 3300 e , 3000 o 2300 16/7 (23) i 2000 V 1 ISOO -U 1000 1 z 300 • 0 -20 - 1 0 0 10 TEUP (OCC C) 4SO0 | 3 S 0 0 ^ io . 3000 • S 2000 • 2. isoo 5 looo u l X 500 0 ' S / 7 (22) 16/7 (11) - 2 0 - 1 0 0 10 TEUP (DEG C) -30 -20 - 1 0 0 TEUP (OEC C) Figure A2b. Temperature (right hand curve) and dewpoint temperature (left hand curve) soundings for Point Mugu during the period 11 UTC July 12 to 23 UTC July 16, 1982. The base of the inversion is indicated by an arrow on each sounding. 190 that the overall characteristics of these weather changes are broadly similar to those described in Sections 3.2 and 3.3 for the May, 1982 case. The differences between these cases is, as analysed in depth in Section 3.3, in the nature of the propagation. During the May, 1982 event, the disturbance moved steadily and progressively northwards as a solitary Kelvin wave until blocked by Cape Mendocino whereas for the July, 1982 case, propagation was unsteady, spurting and typical of a coastally trapped gravity current. Since the nature of the July, 1982 event is well established as a gravity current (Dorman, 1987), attention in this section is devoted towards comparing the observed speed and scales of this disturbance with theory. It is shown that the modified Kubokawa and Hanawa theory of Chapter 2, for the case of weak dispersion, offers a realistic description of this event. As discussed in Chapter 2, if there is significant across-shore flow near and ahead of the leading edge of the intruding gravity current, then dispersive effects are weak and the leading edge steepens to eventually form a shock wave with speed given by (2.58), i.e., 1.45(g'Hu)1/2) w h e r e H u is the gravity current depth behind the leading head. Good agreement is obtained by Kubokawa and Hanawa (1984) between this shock wave theory and data from both rotating tank experiments and observations of the Kyucho coastal current in Japan. A similar theory of shock wave propagation has been derived by Smyth and Holloway (1988) and found to agree well with observations of the internal tide on the northwest Australian continental shelf. Thus, the theory appears to be fairly well tested in a variety of situations. From Figs. A3, A4, it is evident that there are significant across-shore winds near the leading edge of the disturbance. Also, the opposing northerly winds, while not 191 Figure A3. Surface pressure and winds for July 17, 1982 at 1200 UTC. The hatched area denotes the zone of raised marine layer corresponding to the propagating gravity current. Note the across-shore winds just behind the leading edge of the gravity current. Numbers are the surface pressure in millibars minus 1000 and multiplied by 10. Isobars are labelled in millibars. Adapted from Dorman (1987). 192 Figure A4. Surface pressure and winds along the northern Californian coast (just south of Cape Mendocino) on July 17, 1982 at 1800 UTC. The leading edge of the gravity current has reached Point Arena where significant across-shore winds are observed. Wind vectors point downwind. Adapted from Dorman (1987). 193 negligible, are generally weaker than those for the May, 1982 case. Thus, the conditions under which Kubokawa and Hanawa's shock wave theory is appropriate are seen to exist. It will be shown below that (2.58) agrees well with the data unlike the linear Kelvin wave speed used by Dorman (1987). From Dorman (1987), the disturbance along the central and northern Californian coasts was observed to propagate at 8.8 m/s. Application of (3.12) requires the reduced gravity g', which is 0.3 ms from Dorman, and the upstream marine layer depth which can be estimated from the Vandenburg sounding data (Fig. A5) as 450 m. This value is chosen because, according to Kubokawa and Hanawa (1984), H u must be taken as the fluid depth immediately behind the head of the intruding gravity current, or equivalently, as that observed after this leading head has passed; hence for Vandenburg the value of 450 m observed on July 18 after the maximum inversion displacement of 620 m, associated with the passage of this gravity current head on July 17, is appropriate. Thus, the computed shock wave speed is 16.8 m/s. The maximum possible error range is estimated from the data of Dorman (1987) as ±1 m/s. Opposing this is a mean northerly wind of 7.5 m/s (Fig. A6) which then gives a net propagation speed of 9.3 m/s as compared to the observed 8.8 m/s. Dorman however, used the linear Kelvin wave speed which gave 12.0 m/s and hence a net propagation speed of 4.5 m/s, considerably less than that observed. Application of the empirical formula (2.11) for the propagation speed of an atmospheric gravity current of Seitter and Muench (1985) gave a value of only 4.7 m/s and hence also underpredicted the observed speed of the disturbance. It is probable that this lack of agreement is due either to the disturbance being too unsteady in its propagation for formula (2.11) to be valid, or alternatively due to inaccurate determination of the pressure difference 5P, required by (2.11), between Figure A5. Base height of the inversion recorded at coastal sounding stations. Adapted from Dorman (1987). 195 Figure A6. Surface pressure and winds for July 16, 1982 at 1800 UTC. The hatched area denotes the zone of raised marine layer corresponding to the propagating gravity current. Numbers are the surface pressure in millibars minus 1000 and multiplied by 10. Isobars are labelled in millibars. Adapted from Dorman (1987). 196 the gravity current and the ambient atmosphere. The available data (Dorman, 1987) do not have the resolution needed for an accurate computation of 8P. For example, Mass and Albright (1987) showed good agreement between (2.11) and the observed speed of their observed gravity current by calculating 5P from hourly surface pressure data. The shock wave formulation of Kubokawa and Hanawa (1984) is also able to explain, to some extent, another discrepancy noted ,by Dorman (1987), namely that between the observed width of the disturbance and the Rossby radius. Although agreement is satisfactory (within 5 %) near Point Conception and south of 36 N (i.e., Monterey Bay), along the central and northern Californian coasts (e.g., at Point Arena, 38 N) and in southern Oregon (43 N) large discrepancies arise. From Dorman (1987), the observed width at 38 N is 60 km and the Rossby radius is 136 km whereas at 43 N the parameters are 40 km and 123 km respectively. However, using the shock wave formulation, the Rossby radius is computed on the basis of H u and the theoretical width of the disturbance in this case is then less than or equal to 0.7 times this Rossby radius (Kubokawa and Hanawa, 1984; Griffiths, 1986). Rotating tank experiments (Griffiths and Hopfinger, 1983) yielded the constant as 0.6 rather than 0.7. Using H u = 450 m as before then gives theoretical widths at 38 N and 43 N of less than 90 km and 81 km (or 77 km and 69 km if the empirical 0.6 constant is used) respectively. Note that the Kubokawa and Hanawa values are still too large but are nevertheless, considerably closer to the data than Dorman's estimates. The remaining discrepancy could be as a result of using the same estimates of g' and H u for each calculation. Either parameter could vary locally, as a result of changes in the background stratification or the propagation of an upstream frontal wave. Another possibility is 197 that the nonlinearities are sufficiently strong to modify the trapping scale. It has been shown, both experimentally (Maxworthy, 1983) and theoretically (Grimshaw, 1985; Section 2.4), that strong nonlinearities can modify this scale considerably. For example, in some of the rotating tank experiments of Maxworthy (1983) it was observed that the decay scale could be as low as half the Rossby radius. Such a value is certainly in keeping with the observations at 38 N and 43 N. From Section 2.5, the theoretical evolution time for the shock wave is given by (2.53); i.e., by: T = (RH/La)- 5f 1 This formula is applied to data from Oakland which, like Gualala station used in Section 3.4, is on the central Californian coast. From Dorman (1987), R = 143 km while L is estimated as 1521 km (maximum error ±125 km) from the product of the observed propagation speed of 8.8 m/s and time span of propagation of 2 days. To estimate H, the average height of the inversion at Oakland is computed from Fig. A5 to give 480 m. The parameter a, which measures the relative amplitude of the gravity current, is computed as the difference between this height and the climatological mean. This mean is 400 m (Beardsley et al, 1987) and hence a = 80 m. Substitution into (2.53) then gives an evolution time of 54 hours with maximum possible error of ±5 hours. Since the ridge was observed to take 2 days to begin propagating after the initial raising of the marine layer in the Southern California Bight (Dorman, 1987), it is clear that this evolution time for the shock wave is reasonable. Another scale which can be checked with the theory is the rate at which this raising of the marine layer occurred. According to Dorman (1987), the quickest rate observed 198 at Vandenburg was about 400 m over 12 hours. At San Diego and Point Mugu, the soundings (Figs. A2a-b) indicate increases in the marine layer depth over the 12 hours between 00 UTC and 12 00 UTC on July, 16th (when the forcing trough approached the coast) of 219 m and 280 m respectively. This value can then be compared against that predicted by equation (3.4). From Dorman (1987), an estimate of the surface pressure gradient near the formation region (i.e., San Diego) is 1.0 mb/100 km. Substitution into (3.4) then yields an Ekman velocity of 6.6x10"^  m/s and hence a rate of increase in the height of the marine layer of 285 m over 12 hours (with maximum possible error of ±50 m). Theory is therefore seen to agree reasonably with observations despite the fact that the surface pressure rather than that just outside the marine layer (not available) has been used. An unanswered question is why the disturbance ceased propagation at Cape Blanco. Unfortunately, Dorman (1987) makes no mention of this issue in his analysis. Neither the bend in the topography hypothesis nor the increased opposing mean wind speculation seems to offer a complete explanation. In the former instance, the bend in the coastal mountains at Cape Blanco appears somewhat less than that at Cape Mendocino (some 300 km to the south) around which the disturbance had no trouble in propagating. As far as the winds are concerned, Fig. A3 indicates that the winds near Cape Blanco are at most 5 m/s which is significantly less than the observed net propagation speed of 8.8 m/s. Instead, it seems probable that it is the weakening of the reservoir in the Southern Californian Bight which is responsible for the cessation of propagation. Since the disturbance is a gravity current, it requires continual input of the different density fluid to sustain propagation. Fig. A7 shows that the disturbance reached Cape Blanco sometime on July, 19th. From 12 UTC July 17 onwards, soundings (Figs. Figure A7 . GOES-West satellite visible image for 1615 U T C July 19, 1982. The arrow indicates Cape Blanco, which is the northernmost extent of the raised marine layer and stratus overcast. Adapted from Dorman 11987). 200 A2a and A5) at San Diego and Vandenburg show that the marine layer is beginning to lower again, signalling that the reservoir in the Bight is indeed weakening and hence is supplying less cool, dense air into the disturbance. It is probable that it is this reduced supply of dense air to the gravity current that results in the disturbance being too weak to overcome the combined effects of the bend in the topography and the mean northerly wind at Gape Blanco. It is possible too, that the larger scale background synoptic flow has become unfavourable for further propagation. Whatever the exact cause for the cessation of propagation is, there is no intrinsic reason (unlike the May, 1982 solitary Kelvin wave case above) in the dynamics why the disturbance should not have continued propagation other than the argument that it was too weak to do so. Indeed, the more vigorous gravity current event of May, 1985, analysed below, was able to propagate all the way from the Southern Californian Bight to Vancouver Island. To conclude, the event of July 13-20, 1982 has been shown to exhibit all the features of a coastally trapped gravity current. The most important of these, which allow easy identification of the gravity current nature of this disturbance, are the unsteady spurting manner of disturbance, are the unsteady, spurting manner of propagation, the large initial area of raised marine layer in the Southern Californian Bight which acted as an effective reservoir and the good agreement between the observed and theoretical speeds of propagation obtained using the shock wave model. In the next section, the same type of analysis is applied to the more vigorous gravity current case of May 14-17th, 1985. 201 2. California to Vancouver Island event of M a y 14-17, 1985 As indicated by the title, the coastal ridge event of May, 1985 was able to propagate all the way from the initial marine layer disturbance in the Southern Californian Bight to Sartine Island (see Fig. 3.2a for locations of place names) at the northwestern tip of Vancouver Island, B.C., a distance of some 2000 km (Mass and Albright, 1987). Thus, this event was somewhat more energetic than the July, 1982 one and was not significantly affected by coastal irregularities like Capes Mendocino and Blanco and the Juan de Fuca gap. Both weak opposing mean winds (see wind arrows ahead of the leading edge of the ridge - indicated by bold dashed line in Fig. A8) and a large initial displacement of the marine layer at San Diego in the Southern Californian Bight formation region (Fig. A9) are probably responsible for the more vigorous propagation. Satellite images (Fig. A10) and mesoscale pressure maps (Fig. Al l ) show that the disturbance took only 2 days to propagate from San Francisco to Vancouver Island. From an analysis of the changes in the surface wind and pressure fields during the event and a comparison of the observed speed and trapping scale with the Seitter and Muench (1985) formula arid Rossby radius respectively, Mass and Albright (1987) considered this event to be a coastally trapped gravity current. Important details concerning the Mass and Albright analysis are that the computed Rossby radii compared well with the observed width of the ridge whereas for the speeds, the Seitter and Muench formula (2.11) gave a fair approximation to the data. Like the July, 1982 case, it is possible that the discrepancies between the observed speed and that predicted by (2.11) is due to the ridge propagating in an unsteady way. 202 S^^v 1 J ^ ? T ^ 1 ? W l 4 , r e 8 8 U r e a n a l y S 6 S W l n d S f ° r ^ P e r i 0 d 1 2 UTC May 16 to 12 UTC May 17, 1985. The heavy dashed line indicates the leading edge of the gravity current. Adapted from Mass and Albright (1987). C 3 O SAN DIEGO 12 GMT H MAY OO GMT 15 MAY 12 GMT 15 MAY OO GMT 16 MAY 12 GMT 16 MAY Figure A9. Soundings of temperature (solid lines) and dewpoint temperature (dashed lines) for San Diego over the period 12 UTC May 14 to 12 UTC May 16, 1985. Winds (knots) are plotted to the right of each sounding. The base of the inversion is indicated by arrows on each sounding. Adapted from Mass and Albright (1987). 204 Figure A10. GOES-West satellite visible imagery for the period 21 UTC May 15 to 23 U T C May 17, 1985. The stratus deck associated with the propagating coastal ridge is evident. Adapted from Mass and Albright (1987). 205 06GMT 17MAY I2GMT I7M4Y « GMT [7 MAY Figure A l l . Mesoscale sea level pressure analyses for the period 12 UTC May 15 to 18 UTC May 17, 1985. Heavy dashed lines are the northern boundaries of the coastal stratus, as determined from the satellite imagery. Isobars are lOxx mb and observations over the ocean are indicated by solid dots. Adapted from Mass and Albright (1987). 206 Since the interpretation of the May, 1985 event as a trapped gravity current is consistent with the analysis of Chapter 2 and that of Section 3.3 above, attention here will be confined to a more detailed comparison of the observed and theoretical scales of propagation than that provided by Mass and Albright (1987). Forcing for this event consisted of a large synoptic low (500 km or more in diameter) at 500 mb which, as it drifted slowly westward, developed into an open trough. Further details of the synoptic situation and weather changes associated with the propagation of the disturbance may be found in Mass and Albright (1987); it is noted that these are basically similar to those observed during the May, 1982 and July, 1982 cases. As shown for the latter case, the winds near the leading edge of the disturbance (see Figs. A8 and A12) have a significant across-shore component so that the semigeostrophic theory breaks down and it is expected that the gravity current propagates as a shock wave (Kubokawa and Hanawa, 1984) with a speed given by (2.58). This formula is now applied using sounding data from two stations (Oakland, California and Quillayute, Washington - see Fig. 3.2a for locations) to show that good agreement exists between the data and the shock wave theory. Firstly, for Oakland (Fig. A13) the upstream depth H u of the marine layer can be taken as 300 m for 00 UTC May 16 (i.e., after the passage of the leading head of the gravity current). This value has been chosen in exactly the same way as that justified for the July, 1982 case. Unfortunately, Mass and Albright (1987) do not give an explicit value for g' but inferring from their computed Rossby radius gives 0.3 ms"2. Substitution into (2.58) yields a propagation speed of 13.7 m/s (maximum error ±2 m/s) which, after the opposing wind on the central Californian coast of 10 m/s (Fig. A8) has been subtracted, gives a net progression speed of 3.7 m/s. This value compares well with the observations for 00 UTC May 16 of 4 m/s here (Fig. A14). 207 Figure A12. Surface temperature, pressure and winds recorded at Astoria, Oregon during the period 20 UTC May 16 to 12 UTC May 17, 1985. Significant across-shore winds (NE and W) occur particularly near the time (0100 UTC May 17) that the leading edge of the coastal ridge reached this station. Adapted from Mass and Albright (1987). OO o CM OAKLAND 12 GMT 14 MAY 00 GMT 15 MAY 12 GMT 15 MAY 00 GMT 16 MAY 12 GMT 16 MAY Figure A13. Temperature (solid lines) and dewpoint temperature (dashed lines) soundings for Oakland over the period 12 UTC May 14 to 12 UTC May 16, 1985. Winds (knots) are plotted to the right of each sounding. The base of the inversion is indicated by the arrow on each sounding. Adapted from Mass and Albright (1987). 209 Figure A14. Propagation speed of the leading edge of the coastal ridge based on coastal wind changes (solid lines) and the northern boundary of the coastal stratus (crosses). Also shown are the theoretical gravity current speeds calculated from the formula of Seitter and Muench (1985). Adapted from Mass and Albright (1987). -30" -2CT -10* 0* 10* 20* 30*C -10* 0* 10* 20*C -)0» 0* WC 0 0 G M T 17 M A Y 12 G M T 17 M A Y 0 0 G M T 18 M A Y Figure A15. Temperature (solid lines) and dewpoint temperature (dashed lines) soundings at Quillayute, Washington before (00 UTC May 17, 1985) and after (12 UTC May 17, 00 UTC May 18, 1985) the passage of the leading edge of the ridge at that location. Winds (knots) are plotted to the right of each sounding. The base of the inversion is indicated by the arrow on each sounding. Adapted from Mass and Albright (1987). 210 At Quillayute, Fig. A15 indicates that the upstream depth H u is only 250m (data chosen for 00 UTC May, 18 which is after the passage of the leading head of the gravity current). Assuming a value for g' = 0.49 ms"2, which was given as representative for the coastal atmosphere in Washington during the gravity current events of July, 1984 and May, 1985 by Dr. Mass and his group at the University of Washington (Hermann et al., 1989), then gives a propagation speed of 16.0 m/s (maximum possible error ±1 m/s). The opposing mean winds on the Washington coast are at most only 2 m/s (Fig. A16) so the net propagation speed is then 14.0 m/s which agrees very well with the observations (12-14 m/s, Fig. A14). Further support for the shock wave model follows from the similarities between the surface microbarograph traces observed during the event (Fig. A17) and the displacement profiles given by Kubokawa and Hanawa (1984). Next, the theoretical evolution time scale of the shock wave model are compared with those of the disturbance. The mean inversion height during the event at Oakland (Fig. A13) is estimated from the sounding data as 568 m and hence the ratio H/a = 568/168. Assuming g' = 0.3 ms"2 and H = 568 m gives R = 145 km, as compared to 168 km for Vandenburg (some 300 km to the south of Oakland) given by Mass and Albright (1987). The alongshore length scale L is estimated as 864 km from the time span of propagation (2.5 days) and propagation speed along the Californian coast (4 m/s) given by these authors. Hence, from (2.53), the evolution time of the gravity current is evaluated as 52 hours (maximum possible error ±5 hours). Unfortunately, Mass and Albright (1987) give no indication of the scales of the disturbance in its early stages. Sounding data for San Diego where generation occurred (not shown in Fig. A9) indicate that the inversion height was 1700-3500 m, i.e., displaced far above the coastal mountains, for the period 00 UTC 11 May to 00 UTC 13 May. Two days 211 :—i—1 • • *—i—i—i J—i—* • » '—t—i ' • • « • • » • 00 01 0* 01 « l l * 21 00 01 0* W « l ) « 2 I O O O l a O t 8 U * 2100 IS 16 HAT 17 18 nut icun olonjiho't C O A W X M (•. no'ih»cd) c o i l met componcot (» ,Ki lxvd) Figure A16. Surface winds recorded at various coastal stations (see Figs 3.2a-b for locations of place names). Solid dots indicate three hourly observations and dotted lines indicate missing data. Tick marks represent 5 knot intervals. Adapted from Mass and Albright (1987). 212 BAROGRAPH TRACES 1012 i i i i i I I I I T —r — • — O u i H o y u t t . Wo. —^Hoquiom, Wo. j : • Ailorio, Or. . i - Newport, Or. - .Ho-'.'J1. flto0..Or.....: • i i i i t i 1 1 1 1 9 12 IS 18 21 0 3 6 9 12 IS MAY 16 MAY 1? TIME (CMT) Figure A17. Microbarograph traces of surface pressure recorded at various coastal stations from 08 UTC May 16 to 18 UTC May 17, 1985. The vertical arrows indicate times of passage of the leading edge of the coastal ridge. The dashed horizontal lines indicate the 1012 mb pressure level and the tick marks on the vertical axis are at 1 mb intervals. Adapted from Mass and Albright (1987). 213 later, the coastal ridge began propagating along the Californian coast (Fig. All) . From 00 UTC 13 May until the cessation of propagation, the inversion at San Diego was below the mountain crests causing trapping and alongshore propagation. Hence, ah estimate of the likely evolution time for the ridge of 2 days may be made. Further support for the theoretical evolution time of the shock wave is evident in the very recent numerical simulations of Hermann et al (1989) who found that the initial gravity current developed within 10.4 hours of the spin up time and a shock wave at the leading edge within 42 hours. Thus, the theoretical value (52 hours) is reasonable. The soundings at San Diego (Fig. A9) can be compared with the predictions of equation (3.4) for the rate of increase of the marine layer depth. At 850 mb, the pressure gradient near Point Conception associated with the forcing can be estimated from the synoptic maps in Mass and Albright (1987) as 0.75 mb/100 km. Application of (3.4) then gives marine layer depth increases over 12 hours of 125 m (maximum probable error ±30 m). This value is consistent with the observed value of 150 m/12 hours (inferred from the soundings of Fig. A9). Fig. A9 also shows that the winds at San Diego were generally onshore, consistent with the event occurring as a coastal ridge rather than as a trough. As noted earlier, the disturbance was able to propagate as far as the northwestern tip of Vancouver Island. At this point, there is a substantial gap between the mountains of the Island and those of the mainland. This gap is considerably greater than that of the Juan de Fuca strait to the south across which the disturbance was able to propagate. Also, note that the opposing winds along the northwestern coast of Vancouver Island are fairly strong (10 m/s from Fig. A8). These factors as well as the overall synoptic flow probably combine to prevent further propagation of the 214 disturbance. It is also likely that friction is important, because as Mass and Albright (1987) note, the surface pressure rise and wind shifts associated with the disturbance get progressively weaker on the northern Washington and Vancouver Island coasts. In conclusion, the May, 1985 event has been shown by Mass and Albright (1987) and more conclusively by the extended analysis here to be a coastally trapped gravity current which, after an initial evolution time, propagates as the shock wave form of the associated nonlinear coastal wave of Kubokawa and Hanawa (1984). Good agreement was found between observations and theory for the propagation speed, the time scale of evolution and the rate of increase of the initial marine layer depth in the Southern Californian Bight. Very recently (Hermann et al., 1989), a fully nonlinear reduced gravity model of the atmospheric marine layer and the oceanic mixed layer coupled through a parametrized wind stress has been solved numerically for this event to show the oceanic response and that the influence of sea surface temperature variations on the atmospheric gravity current are negligible. The atmospheric part of the model was able to reproduce qualitatively the observed gravity current front profile in both the offshore (curved - e.g., see Fig. A8 and leading edge of stratus in Fig. A10) and the alongshore directions and gave good predictions for the speed and evolution time scale (similar value to the Kubokawa and Hanawa (1984) model used here). However, the model did not give any information as to the generation or evolution processes of the gravity currents nor did it resolve the complicated vertical structure of the leading edge of the gravity current. With the exception of the modelling of the curved offshore profile (and this can be achieved in an approximate way - see Maxworthy, 1983; Grimshaw, 1985) the modified Kubokawa and Hanawa analytic model presented here has been equally successful. It would seem that until computational resources allow efficient three-dimensional treatments of the dynamics of these gravity currents, analytical models provide a practical method for research. 

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