Question: What river's banks was the castle on where Andrew Dudley participated in the War of the Rough Wooing?

Context paragraphs:
Rough Wooing: The Rough Wooing (December 1543 – March 1551) was a war between Scotland and England.  Following its break with Rome, England decided to attack Scotland, partly to destroy the Auld Alliance, and prevent Scotland being used as a springboard for future invasion by France, partly to weaken Scotland, and partly to force Scotland to agree to a marriage alliance between its child Queen Mary and the English heir apparent Edward, son of King Henry VIII.  An invasion of France was also contemplated.  War was declared by Henry in an attempt to force the Scots to agree to a marriage between Edward and the infant queen, thereby creating a new alliance between Scotland and England.  Edward, crowned king in 1547 at the age of nine, continued the war for a time under the direction of the Duke of Somerset before Somerset's removal from power in 1549 and replacement by the Duke of Northumberland, who wished for a less costly foreign policy than his predecessor.  It was the last major conflict between Scotland and England before the Union of the Crowns in 1603, excepting perhaps the English intervention at the Siege of Leith in 1560, and was part of the Anglo-Scottish Wars of the 16th century.

Burning of Edinburgh: The Burning of Edinburgh in 1544 by an English sea-borne army was the first major action of the war of the Rough Wooing.  A Scottish army observed the landing on 3 May 1544 but did not engage with the English force.  The Provost of Edinburgh was compelled to allow the English to sack Leith and Edinburgh.  However, the Scottish artillery within Edinburgh Castle harassed the English forces, who had neither the time nor the resources to besiege the Castle.  The English fleet sailed away loaded with captured goods, and with two ships that had belonged to James V of Scotland.

List of battles between Scotland and England: The Kingdom of England and the Kingdom of Scotland fought dozens of battles with each other.  They fought typically over land, particularly Berwick-Upon-Tweed, and the Anglo-Scottish border frequently changed as a result.  Prior to the establishment of the two kingdoms, in the 10th and 9th centuries, their predecessors, the Northumbrians and the Picts or Dal Riatans, also fought a number of battles.  Major conflicts between the two parties include the Wars of Scottish Independence (1296–1357), and the Rough Wooing (1544–1551), as well as numerous smaller campaigns and individual confrontations.  In 1603, England and Scotland were joined in a "personal union" when King James VI of Scotland succeeded to the throne of England as King James I. War between the two states largely ceased, although the Wars of the Three Kingdoms in the 17th century, and the Jacobite Risings of the 18th century, are sometimes characterised as Anglo-Scottish conflicts, despite really being British civil wars.

Siege of Haddington: The Sieges of Haddington were a series of sieges staged at the Royal Burgh of Haddington, East Lothian, Scotland, as part of the War of the Rough Wooing one of the last Anglo-Scottish Wars.  Following Regent Arran's defeat at the battle of Pinkie Cleugh on Saturday 10 September 1547, he took Haddington, with 5000 troops including French mercenaries and troops sent by Henry II of France to bolster the Auld Alliance.  Afterwards, Francis Talbot, 5th Earl of Shrewsbury took it with nearly 15000.  The English forces built artillery fortifications and were able to withstand an assault by the besieging French and Scots troops supported by heavy cannon in July 1548.  Although the siege was scaled down after this unsuccessful attempt, the English garrison abandoned the town on 19 September 1549, after attrition by Scottish raids at night, sickness, and changing political circumstance.

Ralph Bulmer (soldier): Sir Ralph Bulmer (d.1558) of Wilton in Yorkshire (present day Redcar and Cleveland), was an English knight and soldier active on the Scottish border and during the war of the Rough Wooing.

Fatlips Castle: Fatlips Castle is a pele tower in Roxburghshire, in the Scottish Borders.  Situated at the top of Minto Crags, above the River Teviot, it was built in the 16th century by the Turnbulls of Barnhills, notorious Border reivers, and owners of nearby Barnhills Tower and farm.  In 1545, during the War of the Rough Wooing, the Earl of Hertford burned "Mantoncrake", or Mynto Crag.

Broughty Castle: Broughty Castle is a historic castle on the banks of the river Tay in Broughty Ferry, Dundee, Scotland.  It was completed around 1495, although the site was earlier fortified in 1454 when George Douglas, 4th Earl of Angus received permission to build on the site.  His son Archibald Douglas, 5th Earl of Angus was coerced into ceding the castle to the crown.  The main tower house forming the centre of the castle with four floors was built by Andrew, 2nd Lord Gray who was granted the castle in 1490.

Battle of Pinkie Cleugh: The Battle of Pinkie Cleugh, sometimes known as the Battle of Pinkie, took place on 10 September 1547 on the banks of the River Esk near Musselburgh, Scotland.  The last pitched battle between Scottish and English armies, it was part of the conflict known as the Rough Wooing, and is considered to be the first modern battle in the British Isles.  It was a catastrophic defeat for Scotland, where it became known as Black Saturday.

Andrew Dudley: Sir Andrew Dudley, KG (c. 1507 – 1559) was an English soldier, courtier, and diplomat.  A younger brother of John Dudley, 1st Duke of Northumberland, he served in Henry VIII's navy and obtained court offices under Edward VI.  In 1547–1548 he acted as admiral of the fleet and participated in the War of the Rough Wooing in Scotland, where he commanded the English garrison of Broughty Castle.  He was appointed captain of the fortress of Guînes in the Pale of Calais in late 1551.  There he got involved in a dispute with the Lord Deputy of Calais, which ended only when both men were replaced in October 1552.

William Eure, 1st Baron Eure: William Eure, 1st Baron Eure (c.1483–1548) of Witton was an English knight and soldier active on the Anglo-Scottish border.  Henry VIII of England made him Baron Eure by patent in 1544.  The surname is often written as "Evers".  William was Governor of Berwick upon Tweed in 1539, Commander in the North in 1542, Warden of the East March, and High Sheriff of Durham.  During the Anglo-Scottish war called the Rough Wooing, Eure and his sons Henry and Ralph made numerous raids against towns and farms in the Scottish Borders.

Supporting paragraphs: Andrew Dudley Broughty Castle
Answer: Tay