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HOUSEBUILDER Taylor Wimpey rounded off its refinancing plan yesterday by announcing a new £100m debt facility and plans to sell £250m of five-year bonds.
The news follows on from the firm's £950m credit facility, which was announced last week as the last act by outgoing chief financial officer Chris Rickard.
Prudential's M&G UK Companies Financing Fund will provide the company's £100m facility.
The company has applied for a credit rating from Moody's and Standard & Poor's (S&P). S&P yesterday gave the firm a B+ rating for its bond issue, with a negative outlook due to uncertain market conditions.
"We are likely to lower the rating if the company does not refinance its debt before July 2011 at the latest, or if the company does not manage its debt leverage very carefully while acquiring new land," said S&P analyst Maxime Puget.
The firm rejigged its debt last week in an effort to gain more flexible terms. Chief executive Pete Redfern said its new structure lifted conditions on buying new land.
Shares closed up 3.3 per cent at 25.3p yesterday.
TAYLOR Wimpey's bond issue will be coordinated by Royal Bank of Scotland's corporate broking branch Hoare Govett and Barclays Capital, while Lloyds TSB Corporate Markets and HSBC will act as bookrunners.
Antonia Rowan is a managing director of RBS Hoare Govett, and has been with the firm since 1999 when she joined from Merrill Lynch. She acted as joint broker for Taylor Wimpey's predecessor George Wimpey when it merged with Taylor Woodrow in 2007, and has also worked with Tate & Lyle, Wolseley and Wm Morrison. She is joined by Hoare Govett managing director Iain Leigh-Pollitt.
Marion Dakers


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I had this whole blog post written up that I was working on and I just scrapped all of it. The post was essentially a spillage of my emotions about where I am in my life right now – geographically, mentally, spiritually, and any other "ally" that you may wish to throw in there. After sitting in my drafts folder for a few days, and with a few different ideas in my brain, I realized the post was much too self-pitiful and came to no conclusion. That is not the kind of post I want to include in my blog – that is the kind of thing I need to write in my personal journal.
With an attitude adjustment and some new thoughts, I want to focus on the same issue but with a more positive and productive filter.
In the past few months of being a college graduate, I think the "betwixt and between" phase has left me somewhat confused. This is the first time in two decades that I have not been in school, and the time away from the classroom has sent me straight to Identity Crisis-ville. Who am I, if I am not a student? What do I do now? These are not novel problems; many college graduates go through these thoughts, especially at this juncture in economic history. Novel or not, these issues are important to me and my growth (hopefully) into a "functioning cog in some great machinery," contributing to something greater than myself. How do I go about getting out into the world, outside of my hometown, to serve some kind of purpose as well as find personal happiness?
I've dwelt on the fact for a while now that I am not very well-traveled. I stayed in my hometown for college and am still here working, moving into my own apartment, and making more and more commitments that will keep me here and from being "out there," for a continuing stretch of time. While I am itching to get out there, I have yet to make the leap and actually JUST DO IT. That's all it takes, right? JUST DOING IT. Don't think – just do. I hear that a lot. Well, I really wish that were my approach, but that is not the kind of person I am. I am a planner. I over-plan and over-analyze, so when I make a decision I can be sure it is the "right" one. Unfortunately, I spend so much time planning, analyzing, and organizing, and as I am a pretty indecisive person, I rarely get to the doing stage. I have realized this about myself and this is something that I really want to improve upon. Yes, I can keep the planning, organizing part of myself satisfied, but should let a little of that go and be more willing to throw at least a tiny bit of caution to the wind. Other people do it all the time and come out on the other end quite alright, so I should have confidence to do the same. I don't want to be an idiot about it or anything, I mean, I am in a good amount of student loan debt, so I'm not about to quit my job, leave my dog, and go couch surfing for a year with nothing in my savings account. I've got to find some happy medium.
So instead of feeling sorry for myself because I'm still in MHK (which is really a lovely place to be, and I shouldn't be ungrateful about it for a second), I'm going to accept the commitments I have made at this point, follow them through, and use this time to figure some stuff out. I am in the process of moving into my very own apartment for the first time, and am looking forward to the consistent alone/quiet time I will be getting for the first time in a long time. I will devote this time to really looking deep within and figuring out what my next step could be. I don't want to meticulously plan anything more than is necessary, and not for a little while at least, but I do want to make a decision about what step to take next. Where to go next, even if it is not for very long. I don't want to do any "soul-searching," I want to do some "soul-making." Not just meditate and eat Oreos for a year to figure out what the universe is calling me to next, but to search and search actively to find something for myself about which I can be incredibly passionate.
I want to read, read, read all different kinds of material in an attempt to learn about as many different things as possible to find something that piques my interest. I want to do, do, do (not doo doo) new things in and around my community that I've never done before and that stimulate me in new and interesting ways. I want to continue throwing pottery, as this is something that makes me feel differently than anything else I have ever done, and my new apartment is right behind our local arts center where I can do this as I please. I want to knit and challenge myself to create my own patterns as I go. I want to have people over to listen to records, drink good wine, eat great food, and have interesting discussions about everything. I want to abandon as many glowing rectangles as is possible when I can (so this will not include my 40-hour work week which consists of me staring at two computer monitors side-by-side). I want to take my dog to Arts in the Park and experience the local energy of MHK togetherness. I want to get into the habit of doing yoga every single day. I want to learn about the world and its inhabitants, and right now the best way I know how to do that is to use the resources that are available to me right here in the town where I grew up, with the people I love and know so well.
And after that I want to get rid of all my shit and go somewhere.

subset: commoncrawl


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When I first started this blog, I said it was all for me. I would document little bits of our lives so I could possibly scrapbook them later for my son. Okay, after documenting this moment I've decided it's more my mom, "Gramma". This is for her because she can't be here to see all the small events of our day.
So yesterday afternoon for our snack, I decided to pull out the Sun Chips. They have a limited flavor of Cinnamon. My son and I love cinnamon, probably because all I could stomach for my first trimester was cinnamon sugar toast. I've given him these chips before and he's done really well with them in small pieces. Just as I was walking away from him to get his juice out of the frig, he starts to choke. Ok, this isn't the funny part. He's really choking and coughing to get the pieces out and I come back with the juice to try and make him take a sip. He just wants out of his chair and into my arms, so I pick him up. He slowly starts to calm down. I still have to convince him to take a sip of juice.... he keeps pushing it back to me for me to take a drink, which I do a couple times to show him it's okay.
Once he's calmed down, I see if he wants to have a piece of his graham cracker... he loves graham crackers. He's happy now in mom's arms so he opens his mouth and takes it. I promise the funny part is coming.
After a few seconds of chewing, he sneezes.... all over my shoulder.... graham crackers. Yes, this is what being a mother is all about. Never do I walk out of the house completely clean. I'm sure I have tried milk somewhere on my outfit. But just then, I see a glimmer in my son's eyes as he reaches for the pieces of graham crackers that he just sneezed onto my shoulder. Yes, he ate a piece of sneezed out graham cracker that lays on my shoulder.
I love this kid so much it's sickening.
Sorry, no photo. :)
Thursday, November 15, 2007
4 comments:
- AmyBow said...
I think everything I own looks like it has been crawled across by a snail. Snot is a fun fluid.
- Thursday, November 15, 2007 2:23:00 PM
- Shari said...
Thanks for coming by my blog and for your support.
I just love to see what extra little things I'm "wearing" at the end of my days. He's gorgeous.
- Thursday, November 15, 2007 3:22:00 PM
- Jude said...
THANK YOU....Gramma
- Thursday, November 15, 2007 7:14:00 PM
- Wenderina said...
I'll bring my rain slicker this weekend. See you soon!
- Friday, November 16, 2007 9:34:00 AM
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A health care provider who illegally offered prescriptions for opioids that then have been bought all through the Irish Travelers' Murphy Village space in Aiken County additionally pressured feminine sufferers to have intercourse with him.
That is what a prosecutor instructed a federal courtroom Monday as Dr. Fred Gilliard, 76, a basic apply physician specializing in habit remedy, pleaded responsible to illegally distributing oxycodone, a robust, addictive narcotic used as a painkiller.
To put collectively the case towards Gilliard, federal brokers gathered video, audio and recorded telephone-call proof, in addition to interviewing cooperating witnesses.
"Sir, is there something you disagree with within the authorities's recitation of the details?" U.S. District Court Judge Michelle Childs requested Gilliard in a listening to on the U.S. courthouse in Columbia.
"No, ma'am," answered Gilliard, who had come to courtroom in garb he shortly may alternate for a jail's bounce go well with – darkish slacks, a button-down mild shirt with no tie, and white sneakers.
Gilliard's most up-to-date unlawful actions – he beforehand served time in jail for unlawful distribution of medicine – have been uncovered final 12 months. Federal brokers, investigating a white-collar crime ring within the Travelers' Murphy Village neighborhood, acquired a tip that a physician was dealing opioids to the Travelers.
The physician, whom brokers recognized as Gilliard, was offering Travelers with "prescriptions for opioids and different managed substances which have been then being bought all through Murphy Village," prosecutor Jay Richardson instructed the decide.
According to Richardson, the investigation discovered:
▪ Several sufferers acquired prescriptions, together with for opioids, from Gilliard and bought the medicine in Murphy Village
▪ A "cooperating witness" purchased opioids from considered one of Gilliard's sufferers, who then instructed brokers the physician typically "would proposition her for sexual favors in alternate for prescriptions and that he would contact her in any respect hours of the evening"
▪ Federal brokers reviewed Gilliard's telephone information and located "an irregular sample of contacting his feminine sufferers"
▪ One witness recorded her telephone calls with Gilliard, who was recorded as "requesting sexual favors in alternate for prescriptions"
That witness, who was not recognized, then met with Gilliard at a Waffle House final June, off Exit 5 on I-20 in Aiken County.
At that assembly, Gilliard mentioned he would supply the witness with a prescription for hydrocodone, an opioid, however that the witness needed to have intercourse with him.
That assembly was recorded and have become a part of the proof that satisfied Gilliard to plead responsible.
According to information accounts, Gilliard, a 1968 graduate of the Medical University of Georgia, served a number of years in jail within the late 1990s after a federal jury convicted him of wide-ranging fraud in a Medicare billing scheme. In the 1980s, Gilliard served a number of years in jail after FBI brokers videotaped him dropping off a prescription narcotic to a Savannah lodge.
The drug crime that Gilliard pleaded responsible to Monday carries a most 20-year sentence and a $1 million fantastic.
In a day bond listening to, Gilliard's legal professional, John Delgado of Columbia, begged Magistrate Judge Paige Gossett to let Gilliard keep out of jail, a minimum of till he may switch a handful of remaining sufferers to different docs.
"It can't be mentioned that he's a hazard to anybody besides himself," Delgado instructed Gossett.
In his medical apply, Gilliard has helped drug addicts struggle their habit to opioids and "may be very dedicated to their well-being," Delgado mentioned.
Defendants who've pleaded responsible to critical drug offenses will not be usually eligible for bond, Gossett replied, ruling Gilliard needed to stay in jail till he's sentenced. "My arms are sort of tied on this. I'm required by legislation to detain Dr. Gilliard."
Gilliard, who has a Georgia medical license, has been a physician in Evans, Ga., simply contained in the Georgia state line from South Carolina.
For a number of years, federal and native legislation brokers have been investigating against the law ring run out of the Travelers' Murphy Village.
Some 50 Irish Travelers and their associates have been indicted, pleaded responsible and are awaiting sentencing.
However, the investigation is ongoing.
About 1,400 Irish Travelers, many associated by blood and marriage, stay in and round Murphy Village, an unincorporated space close to North Augusta. The group, identified for its closed methods and scams, is claimed to be descended from Irish immigrants who got here to the world within the 1850s.


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Chestnut Meadows, Mirfield, West Yorkshire
£145,000, 3 bedroom semi-detached house for sale
Sold STC
Key Features
Full Description
MODERN 3 BEDROOM SEMI DETACHED,IDEAL YOUNG FAMILY HOME and positioned into a cul de sac on the outskirts of MIRFIELD which has excellent transport and rail links for commuters. Also walking distance to primary and secondary schools. The property comprises of; entrance lobby, lounge, dining room, conservatory, kitchen, 3 x first floor bedrooms and house bathroom. Externally there is a driveway for off road parking leading to a garage which has been part converted to make a utility room and storage space, the rear being enclosed majority lawned with timber decked patio.
Entrance Hall - Solid wood flooring, staircase to first floor accommodation and radiator.
Lounge - 13'4'' x 11'3'' (4.06m x 3.43m) - Gas fire with tiled inset hearth and wooden surround. Bay window, under stairs storage cupboard, radiator and opening into the dining room.
Dining Room - 8' x 7'6'' (2.44m x 2.29m) - Radiator and sliding patio doors to conservatory.
Conservatory - 9'6'' x 7'9'' (2.90m x 2.36m) - Laminate wooden flooring, electric wall heater and French doors to rear garden.
Kitchen - 8' x 6'9'' (2.44m x 2.06m) - A range of wall and base units with integrated fridge, oven, hob and extractor fan over. Solid wood worktops, inset spot lighting, laminate wooden flooring and radiator.
Landing - Storage cupboard
Bedroom 1 - 9'8'' to robes x 7'11'' (2.95m to robes x 2.41m) - Built in wardrobes, radiator.
Bedroom 2 - 9'6'' x 7'8'' (2.90m x 2.34m) - Radiator.
Bedroom 3 - 6'4'' x 6'5'' (1.93m x 1.96m) - Laminate wooden flooring and radiator.
House Bathroom - 6'4'' x6'4'' (1.93m x 1.93m) - 3 piece white suite, with a low flush wc, pedestal wash hand basin, panelled bath with mixer tap shower over and screen and complimentary tiling, radiator.
Garage / Utility - 17'10'' x 7'10'' (5.44m x 2.39m) - Part converted garage to form part utility room and majority part storage space. Power and lighting with single door access to rear.
Externally - Small buffer garden to the front with block paved driveway in front of the garage and nice sized enclosed rear garden which is majority lawned with timber decked patio area.
Mortgage Advice - At Coubrough & Co, we are happy to recommend James Rudd Mortgage Services as our trusted financial services partner. The company was set up by Jim Rudd in early 2016 and we have partnered with him since the beginning. Jim is a fully whole of market mortgage broker and can access many exclusive deals not available through corporate estate agents. Contact us now for an appointment at one of our offices and let Jim take the stress and hassle away whether you are buying or selling from one of our offices.
Print Property Details
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Call our Birkenshaw Branch today on 01274 470 040
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Monday- Relations, Domain and Range- Classwork: skills practice and practice wksht. 6-1 all
Tuesday- Go over 6-1 worksheets. Discuss Functions Classwork: Skills practice 6-2 #1-6 all and Practice 6-4 all
Wednesday- Quiz over 6-1, 6-2, 6-4 Classwork: update notebooks
Thursday- Slope and Rate of Change Classwork: Skills practice worksheet 7-1 all
Friday- Go over Skills practice 7-1. and Continue discussion about rate of change and slope Classwork: Practice worksheet 7-1


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Nice to read that the great idea of moving Cabinet meetings around the country wasn't as slick as we had imagined.
Firstly, it's been done to boost Gordon's image but, sadly, the trip to Leeds last year cost around £200,000 with policing costing £138,000.
The first one in Birmingham cost around £62,000 says Gordon but he didn't add in the coppers' costs.
The next one is in Liverpool on Thursday so hopefully they might work out who is paying the police bill - and hopefully it's not the council taxpayers.
This is a political gimmick and we shouldn't have to be saddled with any costs.
No-one cares about the Cabinet. They are all remote and self-serving so trying to change peoples' opinions won't do a lot for them.
Tuesday, 6 January 2009
Wee Gordon's pouting costs taxpayers dearly
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Saints Linked With New Record Signing ! If They Stay Up !
Tuesday, 1st May 2018 11:06
According to press reports Saints are planning to revamp their squad and when and if safety is assured they will break their transfer record.
Saints are being linked with a £20 million move for Celta Vigo's Pione Sisto, the wide man came to the club's attention a couple of seasons ago when he played in both games in the Europa League for FC Midtjylland when they knocked Saints out.
Saints monitored the player but chose not to make a move when the player moved from Denmark to Spain to join Celta Vigo in the summer of 2016 for 5 million euros.
Saints are of course hampered in any negotiations at the moment in that what division they will be playing in is unclear and until their fate is sorted they can't plan changes to the playing squad.
But rumours of this sort are commonplace at this time of year as agents look to get their clients into the marketplace and alert club's to the fact that they might be available.
Personally I would rather see us linked to a decent central defender than more attacking players.
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A feeling of déjà vu rose with the sun over Swansea this morning. Another manager out the door, another few days of links, speculation, supposed sightings of candidates spotted outside Rossi's sampling the scampi. But for the first time in recent memory, there is hope in the air.
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rose B.
Lesson Details
rose teaches
- Teaches students:
- Ages 5+
- Teaching since:
- 2005
As an experienced, knowledgeable piano teacher, I have an extensive history of guiding a wide range of students to become more accomplished musicians. I have consistently leveraged fresh ideas and a motivational teaching style to optimize student understanding of playing techniques and composer intentions. My students and colleagues have come to recognize me for my passion, dedication, and ability to effectively translate complex ideas to facilitate student understanding. I focus on advanced placement students; her previous students have competed in prestigious piano competitions and won. They have also preformed in several conservatories and venues nationwide including Carnegie Hall.
1 ReviewWrite a Review
My daughter has been taking lesson at Stage Music Center for 2 years and has made great improvement. Caring & experienced. Defenitly recommend!


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I get off at Redding for a quick pit stop. It's midnight and I'm woozy but my seatmate gets off here so I can stretch out and get some real rest. Another driver gets on to ride back home. We cross into Oregon, still on I-5, while I'm asleep. We got a new driver at Medford and now have three drivers as our Sacramento driver is also staying with the bus till Portland. I wake up at Eugene and feel much better. I was so very tired yesterday. I couldn't really sleep but just wanted to sit there with my eyes closed, but I also wanted to see everything so it was quite a struggle. After a good cup of coffee today I'm ready to go.
We take OR-34 and US-20 into Corvaliis and then OR-34 and OR-99E to Albany. We cross an old metal bridge at Corvallis that is rusted with faded green paint. On one side of the bridge is a wooden sidewalk. The bridge looks like it was made with a very old erector set. I went to sleep in the Sacramento delta and woke up to bucolic farms with pine covered hills and heavily loaded lumber trucks. The sky is cloudy and even at full daylight everyone has their headlights on.
The Albany bus stop is at an Albany transit stop. We pass vineyards and more lumber trucks. Everything is so green. We take OR-22 into Salem and I see the state capital. At Salem they used the lift to put on a guy in a wheelchair. It takes up six seats as they have to shove seats together to make room. I think it's great that people in wheelchairs can also take the bus now. All three drivers were helping but our driver said that it was fairly common anymore. I expect it's used more in the northwest and other places where there are shorter runs, unlike Arizona. This is the first time I've seen it. We also have a blind lady in the seat across from me and the driver is very good about helping her on and getting her settled.
The guy behind me is discussing how his aids is progressing, ok it seems. He's even thinking about going back to work. There are half a dozen small children on the bus. This whole trip there have been more little kids than I've seen before on buses. There are people going into Portland to shop. This seems to be a regular thing for many of the people.
We take US-26 and I-405 into Portland. This is a very large bus station with not that many people in it. I get some breakfast and change clothes and feel so much better. A large cup of coffee in my bottle and I'm ready to take off again. A blind guy is sitting next to me. The retired schoolteacher, across the aisle, has a young african american girl as a seatmate. She's from africa but has lived in the US for years and talks like a native american speaker. We chat all the way to Seattle with stops for napping.
We cross the Columbia River into Washington and I-5 follows the Columbia north till it turns west to the Pacific. Vancouver is a very clean looking town, with upscale leanings. Kelso has a combined train and bus station that are very nice. US-12 joins us for a while till it heads west also to the Olympic Peninsula. The Centralia bus stop is at a Texaco station. We wander through Fort Lewis to a bus stop wih no one there. Tacoma is a very interesting looking town wih old brick buildings and a very unusual looking bridge. There are two tall center posts with what look like steel posts going down to the bridge in a spreading pattern.
Finally we pull into Seattle. I get in line for the Northwestern Lines bus that I'll take to Everett. Our first stop is at Amtrak and the train is late but then we're heading north on I-5 after wandering through downtown Seattle. It looks so much like San Franciso with all the hills. We get in the express lanes that carry you through town and spit you out up north. We only hit rush hour traffic after we leave the express lane and are only half an hour late getting into Everett where my son in waiting for me. We follow WA-526, past the enormous Boeing buildings, and WA-525 to Mukilteo.
It's so good to see him and his wife Lauri. Lauri's sister is visiting so we have a nice family visit and a good seafood meal as we watch the Mukilteo ferry unload and load several times. I keep going on about how green everything is, and it is beautiful, without all the mosquitos you get in the east, though there are some. I am so tired that I fall asleep as soon as my head it the pillow.


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Physician Assistant graduate program gets nod to grow
As the fourth fastest growing profession in the U.S., the demand for physician assistants is expected to grow by 27 percent in the decade leading up to 2016.
Marietta College is helping to meet that need with its Master of Science in Physician Assistant Studies program. Recently the program received approval to increase the number of students accepted into the program.
This fall, the College will accept 30 new students into the program and that number will increase to 36 students in the fall of 2010. The approval came from the Accreditation Review Commission on Education for the Physician Assistant (ARC-PA). Until the fall 2008, the program was certified to accept 22 new students each year.
The graduate program begins with the didactic phase, which students learn basic medical sciences and clinical medicine in the classroom and the lab. Afterward, students undertake a 15-month clinical phase, which includes a Capstone Project and required instruction in various clinical rotations in family medicine, internal medicine, general surgery, pediatrics, emergency medicine, orthopedics, psychiatry/behavioral medicine and one elective clinical setting.
The PA profession is the fourth fastest growing profession and the need to continue to train health care providers is essential to the region, said Dr. Gloria Stewart, chair and director of the PA program. It is the hope of the program that as we educate our students here and throughout the region, we will be helping to provide an increase access to health care.
Stewart was involved in the creation of the PA graduate program at Marietta in 2002. It was the fifth such program established in Ohio and has, since its creation, graduated 96 physician assistants.


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Its been exactly a week since my last update, still have a sore arm and non-working fingers so not been doing anything exciting. Today turned out to be big though.
Set off for Kettering General Hospital this morning to get my stitches out and see what can be done about the fingers. Because there has been no progress I was referred to Leicester Royal Infirmary, they wanted to see me today. This itself caused problems because my dad couldn't get the afternoon off work and my friend who had very kindly driven me to the hospital isn't confident with big roads or cities so didn't want to drive to Leicester, so we ended up having an adventure on the train!
I then spent the afternoon being prodded and poked by various doctors, none of whom seemed to have any idea why my fingers don't work. I was a bit worried when not only were they googling for help they were using wikipedia! More x-rays too! Here's my x-ray...yes I was allowed to take it home, and I know there appears to be a bush growing in my arm!
The verdict of the day was, despite being told previously that I didn't, that I have damaged a nerve so the signal isn't getting through to my fingers to move (slightly bizarre as I still have full sensation!). Made more impressive by the fact that to reach the nerve the brake lever that originally impaled me must have passed several muscles, tendons and nerves without causing any damage whatsoever (hence the confusion as to why I'm broken!).
Because I don't live in Leicester I'm expecting a phone call tomorrow detailing an appointment at my local hospital on Monday. Have been told not to eat anything in the morning as there is a good chance I will be having surgery in the afternoon to repair the nerve.
That said, there are a lot of questions I need answering before I decide if I actually want this operation. I need to know risks of further damage, how likely it is to succeed and the recovery time amongst other things, because after all, as annoying as having some non-working fingers is, its not necessarily worth the operation to fix them!
Will update as usual when I hear anything!


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A theory is presented which provides a model for the appearance of critical layers within the flow below a water wave. The wave propagates over constant depth, with constant (non-zero) vorticity. The mechanism described here involves adjusting the surface-pressure boundary condition; two models are discussed. In the first, the pressure at the surface is controlled (mimicking the movement of a low-pressure region associated with a storm) so that the speed and development of the pressure region ensure the appearance of a critical layer. In the second, the pressure boundary condition is allowed to accommodate the reduction of pressure with altitude, although the effects have to be greatly enhanced for this mechanism to produce a critical layer. These two problems are analysed using formal parameter asymptotics. In the second problem, this leads to a Korteweg–de Vries equation for the surface wave, and then the evolution of appropriate solutions of this equation gives rise to the appearance of a critical layer near the bottom; the corresponding problem at the surface can be formulated but not completely resolved. The appearance of a stagnation point and then a critical layer, either at the surface or the bottom, are discussed; the nature of the flow, and the corresponding streamlines are obtained and some typical flow fields are depicted.
1. Introduction
The existence of a critical layer below a surface wave that moves over a flow with non-zero vorticity—often referred to as a 'shear' flow—is well known. This layer is a region in the neighbourhood of a line (in a two-dimensional configuration) at which the wave speed is equal to the speed in the flow below the surface. The conditions appropriate for the existence of a critical layer, and the structure of this critical layer, have been understood for decades (see Benney & Bergeron [1], Davis [2] and Habermann [3] and references therein). Indeed, mechanisms for the generation of cats' eyes within an existing critical layer have also been proposed [4]. More recently, fundamental work on the nature and properties of steady water waves in the presence of vorticity has considerably advanced our understanding of water waves and the predictions of the Euler equation [5,6]. Further, careful numerical studies have produced detailed streamline patterns and surface profiles for arbitrary amplitude waves over (mainly) constant vorticity [7]. Under suitable conditions, such steady waves may contain a critical layer (with associated closed streamlines) [8–10]. (Although a possible description of the structure of the flow near a critical level involves the inclusion of a thin viscous layer [11,12], the work cited earlier makes use of nonlinearity to give the critical layer its structure.) However, all the aforementioned describe scenarios that either contain, or do not contain, a critical level: these exist, or do not, depending on the choice of vorticity and other parameters—mass flow and total energy—that prescribe the flow. There is no mechanism in any of these configurations that will allow the appearance of a critical layer in a flow that, initially, does not contain one. (Of course, one simple reason for this is that most of the analyses mentioned above are for steady flows.) It is the possibility of generating critical layers where none originally existed that we explore here.
The studies of steady, periodic water waves (of arbitrary amplitude), in the presence of vorticity—at least, on the basis of the classical Euler model—have shown that critical layers are possible for certain vorticity distributions, mass flows and total energy. The general background to these studies can be found in Constantin & Strauss [5], and the presence of a critical layer, and its properties, within this same scenario, are discussed in Ehrnström & Villari [8], Wahlén [9] and Constantin & Varvaruca [10]; see also some of the papers cited in these primary references. Most of this work has considered the case of constant vorticity (which, for linear, long waves does not predict the presence of a critical layer; e.g. [13]), and this is the choice that we make here. We suppose that, initially, the flow has no critical layer. For a critical level to appear, the flow necessarily must be unsteady, this unsteadiness being exhibited by allowing at least one of the flow parameters to vary. There is no natural mechanism for varying the vorticity—the flow is inviscid—and it would be an extreme choice to increase/decrease the mass flow rate (by allowing fluid to be pumped in/out in some fashion). Indeed, in the work of Constantin & Strauss [5] and Ko & Strauss [7], the mass flow rate is a fixed parameter, and the vorticity and energy are taken as variable parameters. The most obvious way to proceed, therefore, is to change the total energy (the total 'head') of the system by adjusting conditions on the surface, e.g. by imposing changes in the pressure (in some fashion) at the surface. This simple manoeuvre will allow a transfer of energy to/from the flow at the surface. So what mechanisms might we reasonably choose?
At the outset of this analysis, we must emphasize that the aim is to produce a (mathematical) model that enables a critical layer to appear in the flow. Of course, we shall endeavour to ensure that any mechanism that we invoke is based on sound physical principles, if that is possible, but ultimately we want to explore what is (reasonably) possible within the framework of the Euler equation. We also add that we aim to provide a simple, deterministic model, without recourse to some representation of collections of statistical data that describe the detailed, but complicated, nature of the motion of the oceans under wind action. Thus, with these restrictions in mind, our first task is to investigate what options are available to us; there would appear to be three obvious ones (in the absence of any viscous stresses).
The simplest option—and, in many ways, the most natural—is to impose pressure changes at the surface, so that the pressure changes in time and, possibly, also in space. (The classical theory of surface gravity waves takes the pressure on the surface to be constant, i.e. atmospheric pressure.) The variation of pressure is precisely how a storm manifests itself: typically, a low-pressure region moves over the surface of the ocean. Such a moving pressure distribution is used, for example, in Akers [14], which provides some analytical details, with numerical results. Thus our first model replaces constant pressure at the surface by the following: the pressure at the surface varies in space and time, and moves over the surface. One important consequence of this model is that the total energy (head), represented by Bernoulli's equation at the surface, is no longer conserved; it is this ingredient that opens the possibility to the appearance of critical levels in the flow.
A familiar property of pressure in hydrostatic equilibrium, in an incompressible fluid, is the linear variation in pressure with height or depth. Our second model uses this phenomenon; however, the pressure changes that we shall need are greatly in excess of those that arise by virtue of the existing density of the air above the water's surface. Nevertheless, the essential physical principle is sound. Thus, in this model, we adjust the pressure condition at the surface so that the pressure is reduced on higher waves, implying that energy is transferred to the system.
A third alternative is particularly relevant to wind-driven forcing, in which the pressure distribution is proportional to the wave slope (e.g. [15–17]). This model represents a positive pressure on one face—the wind blows on this face—and a negative pressure on the back face (corresponding, roughly, to flow separation and a reduced pressure); there is no wind load on a horizontal surface. Although this would seem to be a model worthy of detailed examination—and perhaps it should be in the future—it leads to a description of the wave development that is likely to obscure the features that are of paramount interest in this context. In particular, in the small amplitude, long-wave approximation (which is a reasonable starting point) we obtain an equation for the surface wave, η(ξ,t), which takes the form written in suitable non-dimensional variables (and subscripts denote partial derivatives). Here, we are in the moving frame, ξ=x−ct, over a prescribed 'shear' flow U(z), with β the coefficient associated with the wind-driven forcing. If β=0, then we recover the familiar Burns condition defining the wave speed c (and hence describing the conditions under which a critical level might appear in this approximation) [13,18,19]. If β is non-zero, then this equation implies an exponential structure in ξ, describing an exponential increase in amplitude for one sign of ξ; for small β, this same result obtains, but now with . This effect is not what we seek; rather, we are interested in the situation where the variation occurs in time and applies, more-or-less equally, over a region of the surface and of the waves. The first two models (mentioned above) provide this type of mechanism; so we shall restrict our discussion to just these two cases.
In summary, we shall present a version of the classical theory of water waves, in the presence of an underlying vorticity (and we shall consider only constant vorticity), but with a new pressure boundary condition at the surface. Either the pressure will be prescribed at the surface, but allowed to vary on a suitable time scale, or the pressure provided by the atmosphere will include a reduction in pressure with wave height. In the former case, the pressure is independent of the surface wave (although this pressure will drive the surface wave); in the latter, the pressure at the surface is coupled to the amplitude of the wave. In each case, the intention is to seek, within the conventional framework of formal parameter asymptotics for small amplitude, long waves, conditions for the appearance of critical layers. In particular, we shall address the issue of whether the critical layer (and hence stagnation points in the appropriate frame) appears first at the surface or on the bottom; see the numerical work of Ko & Strauss [7], based on the analysis of Constantin & Strauss [5]. Further, we shall also be able to describe the shape of both the critical lines and the corresponding streamlines; some alternatives, that we can hope to recover, are discussed in the general considerations provided by Ehrnström & Villari [8], Wahlén [9] and Constantin & Varvaruca [10].
2. Governing equations
We consider a two-dimensional flow (x=(x,z)), such that (plane) waves propagate in the positive (or negative) x-direction, over a steady flow with constant vorticity; the fluid is both inviscid and incompressible. This problem is non-dimensionalized in the usual fashion: we introduce the undisturbed depth of water, h0—the bottom is flat and horizontal—and a typical wavelength, λ, of the surface wave. Together with an initial, typical (or average) amplitude, a, of the wave, we have the two familiar parameters of classical water wave theory: ε=a/h0 and δ=h0/λ. The velocity, u, is non-dimensionalized using (with , where → denotes 'replace by'), and the pressure uses ρgh0, where ρ is the constant density of the water and g is the acceleration of gravity. It is then convenient to scale in order to remove δ from the equations, by writing , and , so that the problem is now formulated for arbitrary δ. Thus, we obtain the set 2.1 with 2.2 and 2.3 where the surface is written as z=1+εη(x,t;ε) and p is the deviation of the pressure away from the hydrostatic pressure distribution. The term P(x,t;ε) represents the chosen pressure condition at the surface, relative to the constant atmospheric pressure. (Note that the condition p=εη on z=1+εη, i.e. P=0, is simply a result of measuring p relative to the hydrostatic pressure distribution, with pressure=atmospheric pressure=constant on the surface.)
The final form of the equations, as we shall work with them, is obtained by introducing a suitable frame associated with the wave, together with a suitable time scale: 2.4 respectively, where c(τ;ε) is a variable wave speed. Further, the problem of interest here requires the choice 2.5 and the underlying (Galilean) frame is chosen so that the undisturbed flow (u=−γz,w=0) is zero on the bottom. The vorticity in this background flow is simply 2.6 the sign being chosen to correspond to the convention used in much of the earlier work (which was based on (x,y) coordinates); we are not concerned, here, with the case of irrotational flow, γ=0. Now equations (2.1)–(2.3), with (2.4) and (2.5), become 2.7 2.8 and 2.9 with 2.10 2.11 and 2.12 The vorticity associated with the perturbation away from the background state is then We seek a solution in 0≤z≤1+εη(ξ,τ;ε), for and τ≥0, which possesses an asymptotic structure based on ε→0. The two models that we now examine are
model 1: pressure P(ξ,τ) is prescribed;
model 2: set P=−αη(ξ,τ;ε) for a given constant α>0.
3. Model 1: formulation
This choice involves a prescribed pressure on the surface that moves (at a given but, in general, variable speed) and which may also evolve on the slow time scale. An immediate issue is the relation between this speed and the speed of propagation of the surface waves in the absence of any forcing; this must be addressed before we can proceed. In order to do this, we briefly revert to the equations written in terms of the original (non-dimensional, scaled) x and t, with ε→0. From equations (2.7)–(2.12), we obtain the set with Directly, we find that the equation for η(x,t), at this leading order, is and so we obtain where where c1, c2 are the roots of C2+γC−1=0.
Now with and c≠c1,c2, the solution for η is for arbitrary functions F(⋅) and G(⋅), given P and c(τ). On the other hand, if c=c1 (or, correspondingly, c2), then the particular integral for η becomes but, as we now discuss, this will not be relevant to the development that we present here.
We suppose, in the near-field (defined by t=O(1) and containing the time, e.g. t=0, at which initial data are prescribed), that we have suitable initial data on compact support (over a domain x=O(1)). In general, there will be three wave components: one moving upstream and one downstream (determined by the speeds c1, c2), and the one associated with c. Unless c is very close to either c1 or c2, then when the three components have reached the far-field (t=O(ε−1) or τ=O(1)) they will be distinct and separate waves, no longer overlapping and, indeed, moving further apart. Now, we want to vary c(τ) so that criticality occurs; this requires c to attain a value of 0 or −γ, and then to decrease/increase further to reach a value between 0 and −γ. In order to avoid the difficulties of c passing through the values c1 or c2, we elect to discuss the problem in which c(τ) starts from a value that satisfies 3.1 and 3.2 and not close to either c1 or c2. Thus, in the far-field, we have only one component to consider: the one moving at speed c, which is driven by the moving pressure distribution. (We should remember that the aim in this study is to develop a mathematical model that describes the appearance of a critical layer, not to analyse all aspects of a more general problem—which might be a worthy subject for a later study.) Consequently, we return to equations (2.7)–(2.12), which are valid in the appropriate far-field, and seek an asymptotic representation of the surface wave, η(ξ,τ;ε), and of the flow field.
To proceed, we assume that there exists a (formal) asymptotic solution of (2.7)–(2.12), which takes the form 3.3 and 3.4 for given P(ξ,τ) and c(τ). We shall assume throughout that the choice of c(τ), and particularly of P(ξ,τ), will allow the construction of uniform expansions. A cursory examination of the structure of this problem suggests that it is sufficient for c(τ) to be a C0 function, but we should allow P(ξ,τ) to be . (We are at liberty to choose both these functions, so we may be as prescriptive as necessary; we comment further on the uniform validity of this expansion later.) The solution, at leading order, follows directly as but since c2+γc−1<0 for our choice of c(τ), it is convenient to write 3.5 This is consistent with the scenario that we have constructed: a wave of elevation is produced by a reduced pressure (P<0) on the surface. (Similar issues, with c inside the domain (c1, c2)—or (c2,c1), depending on the sign of γ—are mentioned in Akers [14].) There is no need, at this stage of the analysis, to consider higher order terms; we shall do that later.
Thus, we have 3.6 with 3.7 and so stagnation points (u−c=0 in this frame), and then critical levels, can appear if c+γ or c are sufficiently small. Note that we use here the full, original (non-dimensional) expression for u: background flow+perturbation. The form of this asymptotic solution suggests that a critical size will be c+γ=O(ε) or c=O(ε)—and remember that, in this model, we may choose c(τ). Let us set 3.8 or 3.9 and so a stagnation point can occur first at z=0 (the bottom) for (3.8), and at z=1 (the surface, in this approximation) for (3.9). However, we require a choice that, initially, ensures that u−c<0 (which guarantees no stagnation within the flow field) but which allows u−c to become zero and then positive, as the pressure distribution moves and evolves. Of the various options available to us, we limit ourselves to waves of elevation (so −P>0; see (3.5)), and then for (3.8) 3.10 and for (3.9) 3.11 The possibility of stagnation points appearing first at the bottom or on the surface (and along the vertical line through the peak, as we shall also see) is already known [7,20–22]. Of course, our asymptotic approach—based on small amplitude, long waves—cannot predict a critical value of γ that separates these two regimes (as was found in the numerical work of Ko & Strauss [7,23], for steady, periodic waves).
Now that we have seen how our initial investigations have predicted the possibility of stagnation points and eventually (we expect) critical levels, we incorporate the choices (3.10) and then (3.11) into the full equations, and analyse more carefully.
4. Model 1 with c=εC(τ)
The procedure here is to write c=εC(τ) in equations (2.7)–(2.12) and, in addition, to observe that this choice implies p=O(ε) (see (3.7)) and so we also write p→εp. The governing equations in this case are therefore 4.1 4.2 and 4.3 with 4.4 4.5 and 4.6 Further, in order to take the asymptotic expansion to higher order, we employ the familiar technique in these problems that involves expanding the surface boundary conditions about z=1. (This is easily confirmed as a valid procedure here, in common with all long-wave approximations in water wave theory, because the solution is polynomial in z.) We seek an asymptotic solution of this set, (4.1)–(4.6), using the asymptotic expansion introduced earlier (see (3.3) and (3.4)); at leading order, this gives (for γ≠0) with The solution, for which any disturbance is driven solely by P, takes the form The extension to the next order is routine, producing (for example) where corresponding results are available for p1 and η1. It is clear that these terms, together with the higher order terms, constitute a representation that is uniformly valid, provided that P and all its derivatives—each term 'n' involves 2n+1 or 2n derivatives in ξ (depending on the function being expanded)—remain bounded. It is typical of far-field expansions for wave propagation, that validity can be demonstrated, for suitable initial data, for times τ∈[0,τ0] (equivalently t∈[0,τ0/ε]) for some fixed τ0 as ε→0.
Let us choose to consider the situation in which P moves according to c=εC(τ), but does not evolve in τ, i.e. P=P(ξ); then On this basis, we construct an asymptotic representation of u−c: The critical level (where u−c=0) is then determined by 4.7 to leading order, or 4.8 when the next order is included. If this prescription implies z<0, then there is no critical layer within the flow field; if z=0, then a stagnation point (or, possibly, stagnation line) appears on the bottom; if z>0, then a critical layer sits in the flow field. In order to describe, in simple terms, what this predicts for the flow, let us consider a function −P(ξ)>0 with a single peak at ξ=0, and which otherwise decays to zero as (or, essentially equivalently, is on compact support on [−ξ0,ξ0] for some ξ0>0).
For γ>0, then (to leading order) if there is no stagnation point, nor a critical layer within the flow; when C(τ) is reduced to , a stagnation point appears on the bottom, precisely below the peak of the surface wave (because η∼−P; figure 1a). Now, as C is reduced further, so that , there is a line given by (4.7)—the critical level for this flow—that follows the shape of the surface and extends between two stagnation points on the bottom. Once a critical line has formed inside the flow field, closed streamlines then arise. The streamlines, in this moving frame, are determined from to this order. Thus, we obtain an expression for the streamlines: 4.9 which necessarily sit in an O(ε) neighbourhood of the bottom, by virtue of the approximations used here (although we can readily reposition this region by moving the critical line further into the flow). We comment that it is easily confirmed that there is not a separate asymptotic region (a boundary layer) near the bottom; the solution constructed here is uniformly valid from top to bottom, across the flow field. All this information is depicted in figure 1b. In these two cases, the wave is moving (slowly) to the right. If, eventually, C<0 (so the wave has now changed direction), a critical level extends to infinity in each direction (at least, for our chosen form of –P; figure 1c). (If we include the higher order contribution given in (4.8), then the values of C just described are adjusted by O(ε)—the changes depending on γ, and P2 and Pξξ at the peak—but the overall picture is unaltered.) A similar sequence emerges in the case of γ<0 with C<0 initially: —no stagnation points; —there is a single stagnation point on z=0 directly below the peak; —a critical level exists within the flow field.
5. Model 1 with c=−γ+εC(τ)
The development here follows the previous one quite closely, although some of the details are, in important areas, different. First we note that the choice c+γ=εC(τ) implies that, not only is p=O(ε), but so are u and w. Thus, we perform the transformations and then equations (2.7)–(2.12) become with The same asymptotic structure as used previously then gives where Again, let us allow C(τ) to evolve but choose P=P(ξ), then 5.1 thus, at this order, we obtain 5.2 Then, for u−c<0 near z=1 with C=O(1), we require C>0 and so γ<0 (as noted in (3.10)). When we include the detailed shape of the surface (z∼1+εη0+ε2η1), we see that 5.3 At leading order, (u−c)|surface<0 if ; a critical layer will appear (near the surface) if , and a stagnation point first arises at the peak when . The critical level (where u−c=0) is defined by the curve which is independent of the shape of the free surface; cf. the critical level near z=0. Further, we can easily obtain the equation of the streamlines (in the moving frame, to this order) in the form 5.4 cf. the streamlines near the bottom, (4.9).
The picture here is therefore somewhat simpler than for that near the bottom: as the surface wave slows down (C decreases), so first a stagnation point appears at the peak and then a line (parallel to the bottom) becomes the critical level; figure 2. (If C decreases further, the critical line will eventually sit below the minimum of the surface and so will extend to infinity in each direction.) The higher order terms merely provide a small adjustment to the shape of the critical line, but this does now involve the shape of the surface. Thus, at leading order, we have predicted the appearance of a critical level near the surface but, importantly, the shape of this line does not follow that of the surface, although—of course—the streamlines within this region do (figure 2).
6. Model 2: formulation
In this model, we replace the arbitrarily assigned pressure (P) by −αη(ξ,τ;ε), where α is a positive constant. We could allow α=α(ξ,τ), but this would not constitute a physically reasonable option: the underlying model here makes use of the hydrostatic pressure distribution in which α is proportional to the density of an incompressible fluid. Here, the only relaxation of reality is to permit this constant to be as large as O(1), if this is required in order to accomplish our aim of generating a critical layer. As we found in model 1, we should first examine, in outline, the general effect of this new surface boundary condition; once we have some guidance, we can make specific choices and analyse the problem more carefully.
The governing equations, following directly from (2.7)–(2.12), with P replaced by −αη in equation (2.11), are 6.1 6.2 and 6.3 with 6.4 6.5 and 6.6 One further adjustment is that, here, we may take ξ=x−ct, where c is a constant to be determined (but which may depend on ε); c is the speed of linear, long waves in this flow. We therefore tackle the problem in a way similar to that which produces the classical Korteweg–de Vries (KdV) equation in water wave theory, although, as we shall see, the balance of terms is rather different here.
We invoke the same asymptotic structure as in our previous discussions (see (3.3) and (3.4)), together with ; at leading order we then obtain (from (6.1)–(6.6)) for arbitrary η0(ξ,τ), this solution being defined for z∈[0,1]. In this derivation, we have assumed—as in our earlier work—that the only disturbance to the background state is that generated by the surface wave η∼η0.
The possible speeds of propagation are given by and these two roots give speeds that are either faster or slower, as appropriate, than the maximum and minimum values of the constant vorticity background flow (precisely in accordance with the Burns condition [13,19]). Nevertheless, we may get close to these maximum/minimum values (0,−γ) by allowing 1−α→0. Indeed, we see that, with 1−α=Δ→0, then 6.7 In this exercise, we seek a choice of (the size of) the constant Δ that will allow, as the wave evolves, the appearance of a critical layer. In the first option given for u−c, we see that a critical level might appear near z=0, as η0 increases, if Δ=O(ε) and γ>0. (We shall always ensure that 1−α=Δ>0 in order to maintain some contact with physical reality.) The second expression for u−c suggests that a critical layer, if it appears at all, will be near the surface. In this case, however, the situation is less clear: with z=1 (the surface, to leading order), the term associated with the surface wave (εΔη0/γ) is always smaller than Δ/γ, and so it is not immediately obvious how a balance is possible that will change the sign of u−c. On the other hand, the surface is more accurately z∼1+εη0, so a way forward may become evident; we shall show that the problem can be formulated, but it gives rise to a more general variant of the KdV equation (which we cannot solve).
Before we proceed, we should comment that the equations (6.1)–(6.6) possess a scaling property that leads to an alternative interpretation of our approximation. The scaling p→(1−α)p, , , (and then removes the factor 1−α from the problem; however, this is at the expense of changing the size of the vorticity as Δ→0—and we wish to retain γ=O(1)—and, more significantly, of changing the time scale. This latter result is not consistent with the underlying approach that mirrors the classical conditions and time scales that lead to the KdV equation, which is what we hope for. The scaling property does imply, however, that this same analysis can be recovered by a suitable limit of the constant vorticity, with an evolution of the flow on a suitable time scale. Here, we choose to retain, as a separate parameter to be adjusted, 1−α=Δ, with the time scale defined solely in terms of the initial amplitude of the wave, and the vorticity fixed as an O(1) constant.
In what follows, we set 1−α=Δ=ελ (λ=O(1)), and then consider the two available choices for c, the first being c=O(ε); so we write c=εC.
7. Model 2 with c=εC
The boundary condition (6.5), with 1−α=ελ, implies that p=O(ε), and so we transform according to p→εp; equations (6.1)–(6.6) then become with We seek a solution based on our familiar asymptotic expansion (see (3.3) and (3.4)) which gives, at leading order (valid for z∈[0,1]) for arbitrary η0(ξ,τ) (and C0 is also undetermined at this order). At the next order, after employing—as before—Taylor expansions about z=1 in the surface boundary conditions, we find that together with a corresponding expression for p1, and with 7.1 and 7.2 We recognize equation (7.1) as the famous KdV equation, written in these far-field variables (ξ=x−εCt,τ=εt), here associated with a contribution from the constant vorticity (γ) of the background flow. This form of KdV equation is consistent with that obtained from the consideration of propagation over a flow with arbitrary vorticity [24]. As we have mentioned earlier, these expressions, and all those that follow this development, constitute terms in a uniformly valid asymptotic expansion (at least for τ∈[0,τ0]), as ε→0, provided that η and all its derivatives remain bounded. This will certainly be the case if the initial data are suitably differentiable.
The conditions for the appearance of a critical layer require (at this order) to be negative, then zero and eventually becoming positive. (We note that this expression for u−c corresponds precisely with that given in (6.7).) It is clear—again, as we have already noted—that any critical layer based on this expression will emanate from a stagnation point on the bottom (z=0). Now u−c<0 on z=0 if γ>0 (because ελ>0) and γη0 is small enough, i.e. (and we shall restrict ourselves to waves of elevation); thus the following description applies only to positive vorticity. The critical level is given (at this order) by provided that this produces Z>0; a stagnation point appears when η0 attains the value λ/γ2, and this will be a single point if η0 possesses an isolated maximum. When a critical level has appeared, the closed streamlines can be recovered from 7.3 at this order; cf. (4.9) and (5.4). So what mechanism will allow the appearance of a stagnation point and then a critical layer?
The simplest scenario is to consider a suitable initial-value problem for the KdV equation (which could be the corresponding initial data in the near-field, which then propagates, essentially unchanged, until it reaches the far-field). This is most easily accomplished by selecting an initial profile—such as a suitable sech2 function—that evolves into exactly two solitons. (This same device was used in Johnson [4] to describe the formation of a cat's eye within an existing critical layer.) Because the amplitudes of the two solitons necessarily satisfy the condition that one has an amplitude greater than the initial profile, and the other smaller [25] we can arrange the following. Let the initial amplitude be less than λ/γ2: there is no stagnation point (or line) within the flow. As the leading—the larger—soliton develops, there will be a time at which the maximum value (the peak) is equal to λ/γ2: a stagnation point appears (at this instant) on z=0, directly below the peak of the soliton. Thereafter, the soliton grows to its maximum amplitude and continues to propagate: there is a critical layer, with closed streamlines—a cat's eye—between points on z=0, which sit directly below those points on the surface at which η0=λ/γ2. For finite time, the surface shape in the neighbourhood of a soliton and the corresponding critical line are not quite symmetric about a vertical axis, but symmetry is approached as the wave moves off to infinity. The resulting flow configuration under this soliton is depicted in figure 3. We add that, for an initial profile that generates more solitons, it is possible to have more than one cat's eye, but these are separated from each other and move apart, because the amplitude between solitons tends to zero as they propagate forwards at different speeds.
8. Model 2 with c=−γ−εC
With 1−α=ελ and c=−γ−εC, we require (as we did in model 1 with the choice appropriate for a surface critical layer; see §5) the scaling to give the equations with The vorticity associated with the perturbation of the background state is then uz−εwξ=0. A routine asymptotic expansion, following exactly the pattern described earlier, yields (for example) 8.1 and 8.2 At leading order, we also find that C0=λ/γ and 8.3 and this equation, which describes the surface to leading order, indicates the presence of a serious difficulty. In this approximation, any surface wave with a negative slope moving to the right (correspondingly: positive/left)—and a typical wave of elevation will certainly possess this property—will progressively steepen, tending towards the vertical at some point on the profile. Of course, we can expect this to be balanced by a contribution from dispersion in the neighbourhood of this steepening front. Indeed, the higher derivative terms in the expansions (8.1) and (8.2) are evident (and it is easy to obtain the pattern for all higher order terms in this respect), and these will become larger as the wave steepens. However, there is an additional complication: we expect such terms to play a role in the equation that describes the surface, and this requires evaluation of w on z=1 (being the surface after the use of Taylor expansions in the surface boundary conditions)—but then this higher derivative term vanishes in (8.1). These observations suggest that the structure of the problem is quite involved.
It is an instructive exercise to explore how the dispersive property depends on depth, driven by the relevant (variable) scaling: 8.4 where (ξ0,τ0) is the position and time, respectively, at which the slope of the profile first becomes vertical (according to (8.3)). The neighbourhood of the surface (n=1), and the bottom (n=0, so z−1=O(1) contains z=0), must be treated separately, the expansions then being matched. We adopt a more direct route that picks out the surface wave (but gives less information about the behaviour-with-depth close to the steepening front); this approach makes full use of the matching principle. We present the main results in outline only because, as we shall see, there is a limit to what we can describe in any detail.
The first stage is to observe that the asymptotic expansion for u, (8.2), for any z∈[0,1+εη], is not uniformly valid where , this scaling corresponding to the general version in (8.4), evaluated for the region away from the surface, i.e. n=0. We introduce With this scaling, we obtain the familiar Laplace's equation for , here unscaled: and so the relevant solution (satisfying the bottom boundary condition) takes the form 8.5 where we have assumed that the appropriate Fourier transform exists; correspondingly, 8.6 both valid as ε→0. This solution, at leading order, incorporates full linear dispersion (in that there is no further approximation involving, for example, long waves). The kinematic condition at the surface requires that to leading order, which then defines A in terms of η.
The solution just described is defined for all z∈[0,1+εη], and so it can be evaluated in the neighbourhood of the surface (with z=1+εZ). However, a careful examination of the form of this expansion—the higher order terms and the form of the governing equations make this clear—shows that it exhibits a further breakdown even closer to (ξ0,τ0): this occurs for (χ,σ)=O(ε) where . Thus we now introduce and then we recover the full, governing equations, bereft of any dependence on the parameter: 8.7 together with the surface boundary conditions 8.8 where we have written η(ξ,τ;ε)≡H(X,T;ε); the relevant vorticity is UZ−WX=0.
At first sight, it might be thought that we have a problem that cannot be usefully addressed. However, we have one significant advantage here: we know the asymptotic structure (and can find a solution to any desired order) for any z∈[0,1+εη], but for a slightly larger (ξ−ξ0,τ−τ0); viz. (8.5) and (8.6). There is now a standard procedure available to us; we solve (8.7), with the boundary conditions (8.8), to produce a solution that is valid near the surface. Of course, this does not generate a solution away from the region of the surface, because it is valid only for Z=O(1). However, we have a solution for all z (and so for Z=O(1), in particular), but valid a little further away from the region of maximum steepness, i.e. (8.5) and (8.6). We match these two solutions, for Z=O(1), thereby ensuring that the solution very close to (ξ0,τ0) contains information from a region further from (ξ0,τ0) which does satisfy the bottom boundary condition; this is a familiar role of the matching principle.
From equations (8.7), with the irrotationality of the velocity field (U,W), we have and so we write 8.9 where β is a constant chosen to ensure that the appropriate integrals exist (e.g. ). (These terms in β can be reabsorbed into B and D, if required.) Correspondingly, we then have 8.10 and we may obtain p directly; the kinematic condition at the surface becomes There is a corresponding equation associated with the pressure condition at the surface (which, of course, we assume is consistent with the equation above—which is certainly the case if a solution exists). The procedure now involves matching the solution represented by (8.9) and (8.10), which is valid for Z=O(1) and , to the solution given by (8.5) and (8.6) (which is valid for , after evaluation on Z=O(1)).
The resulting calculation is altogether routine, but rather intricate; the complete solution in each region, to leading order, matches precisely with the choices and which shows the appropriate transformation between the integrals; the relabelling here, between A and B, is equivalent to that between η and H (used earlier). The essential argument that underpins this process is that we assume that the dominant contributions to the various Fourier transforms come, in all cases, from wavenumbers of O(1). This principle has been applied, most particularly, after employing the appropriate scalings (in terms of ε) between the integrals, followed by ε→0. Further, when we note, at leading order as ε→0, that we can express the equation of the surface in the form 8.11 (The H associated with the hyperbolic functions is evaluated as H(X,T); the integrals in κ can be evaluated in terms of transcendental functions, but there is no virtue in pursuing this here.) This equation incorporates a contribution that is typically associated with linear dispersion—the term on the left—but here it involves full (linear) dispersion and is nonlinear! Correspondingly, the term on the extreme right is a contribution from dispersion to the familiar nonlinear dispersion (as appears, for example, in the Camassa–Holm equation).
It is clear that we cannot continue with our primary aim here: to describe the appearance of a critical layer, near the surface, as the surface wave evolves. We are unable to solve the governing equation, (8.11), for the surface wave, and a detailed discussion of this new equation—no doubt of some interest—is beyond the scope of this current investigation. We can reasonably suppose that solutions exist, analogous to those of the KdV equation, which describe an evolving wave; a suitable solution could then be used (as in §7) to provide a mechanism for the appearance of a critical layer at the surface. This is not a route that we choose to follow, at present.
9. Discussion
Our intention in this paper has been to present mechanisms for the appearance of critical levels in the flow (necessarily with vorticity) below a surface wave. We have seen that, even in the simplest case of constant vorticity, two possible mechanisms are available: one involves no more than the adjustment, in any desired fashion, of the pressure at the surface; the other allows the pressure in the atmosphere to reduce with height. The technique employed (familiar in classical water wave calculations) involves the construction of asymptotic expansions based on a small parameter. In this discussion, we have worked with just a small amplitude parameter—the (arbitrary) wavelength parameter has been scaled out—and the pressure boundary condition is then chosen appropriately.
The first model, which is the most natural one, allows the choice of surface pressure (in a manner that mimics the movement of a low-pressure region over the ocean). The form of this pressure distribution and, more particularly, the speed at which it moves enable a stagnation point, and then a critical level, to appear in the flow. This can be engineered to occur either at the surface or at the bottom, consistent with the general results obtained by Constantin & Strauss [5], Constantin & Escher [26] and Constantin et al. [27], and depicted in the numerical work of Ko & Strauss [7], albeit for steady, periodic waves. Our results produce flow fields of the type predicted by Ehrnström & Villari [8], Wahlén [9] and Constantin & Varvaruca [10]: a critical level, with associated streamlines (cats' eyes), either close to the surface or close to the bottom. In the case of a pressure (–P) with a single maximum, both these flows emanate from a stagnation point either at the peak of the wave, or at the point on the bottom directly below the peak, respectively. Of course, our approach—perforce—is based on a small amplitude approximation (which is the case discussed in Wahlén [9]), and so we are unable to apply our description to large amplitude waves (which are allowed in Constantin & Strauss [5] and Constantin & Escher [26]). However, we can permit the critical level to move away from the region of the corresponding boundary, and into the main body of the flow, by suitably adjusting the speed of the pressure disturbance. The critical layer structure, in this case, will still be constrained to an O(ε) thickness about the critical level, by virtue of the surface wave of amplitude O(ε).
One interesting feature of this description of the critical level, not apparent in the earlier, more general discussions cited above, is that the shape of the critical line is driven by the shape of the other boundary. That is, the critical layer near the bottom follows the shape of the surface wave, and correspondingly for the level near the surface. The shape of the streamlines that describe the critical layer (and represents a cat's eye), in both cases, follows the shape of the surface wave. Finally, we comment that our results predict the appearance of stagnation points, and then critical levels, at the bottom for both γ>0 and γ<0, but only for γ<0 at the surface; these results are in general agreement with those observed by Ko & Strauss [7], but these apply, as we have noted earlier, to steady, periodic waves of arbitrary amplitude.
The second model is, we submit, technically more interesting and, as we have seen, rather more challenging. We now allow the pressure at the surface to decrease linearly with height, so that higher waves are subjected to a lower external pressure. This is clearly what happens in the standard model of an incompressible gas (air) above the fluid interface, but the effects must be greatly enhanced to produce the development of the flow that we seek. Nevertheless, we have a viable mathematical model—our primary aim—that automatically accommodates the desired phenomenon within our governing equations (rather than being arbitrarily adjusted, as in model 1). In this case, the plan has been to develop a suitable KdV-type analysis and then to allow the critical level to appear as a specific surface wave naturally evolves.
We have demonstrated that this procedure works well for the description of a critical level near the bottom, but only for γ>0; no corresponding solution exists for γ<0. The appearance of a stagnation point, and then a critical level (and associated single cat's eye), can be obtained by invoking the two-soliton solution of the KdV equation. The resulting appearance and evolution of the critical level, and of the corresponding streamlines, follow closely that seen in model 1; indeed, the shape-following property described for model 1 is reproduced for model 2. However, the corresponding problem for the appearance of a stagnation point and critical level, at the surface, is far less straightforward.
The problem of a KdV-type description of the development of a critical layer at the surface has not been satisfactorily completed. The asymptotic structure first entails a description that involves the steepening of a wave front (represented by the solution of ητ−γηηξ=0), which then leads to the need for a thin, scaled region close to this front. This in turn requires an analysis of the full, governing equations without the advantage of any small parameters. Fortunately, an asymptotic solution (valid throughout the depth) in an intermediate region fairly close to the front is available; the matching principle has been invoked to obtain a solution near the surface and close to the front. Sadly, as we have seen, the equation describing the surface wave is highly nonlinear—far more so that the KdV equation—incorporating very strong dispersive contributions. This equation cannot be solved; so our discussion of the appearance of a critical layer at the surface must end here. Suffice it to record that there may be some virtue in analysing this new equation—perhaps some additional modelling assumptions could be used (e.g. restricting the wavenumber, κ, in (8.11), or making choices for β)—but this is altogether outside the remit that we have set ourselves in this discussion.
In conclusion, we have introduced two models that admit the appearance, and development, of stagnation points and critical levels in a constant vorticity flow; no doubt the techniques could be applied to flows with more general vorticity distributions (but we should not expect any significantly different conclusions). Some details of the structure, albeit in the asymptotic limit ε→0, have been obtained and described. It has not been possible to complete the discussion of one of the four scenarios that we have introduced.
The author is pleased to acknowledge that both referees indicated a few additional, relevant references that have now been included.
One contribution of 13 to a Theme Issue 'Nonlinear water waves'.
- This journal is © 2012 The Royal Society


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