Conformal mapping of the Gulf of Gdañsk onto a canonical domain

• Main Authors: Pawe³ P. Czesnik, W³odzimierz J. Prosnak
• Format: Article
• Language: English
• Published: Elsevier 2002-06-01
• Series: Oceanologia
• Subjects: Shoreline; Conformal mapping; Domain of solution; Computational domain; Inversion; <i>Schwarz-Christoffel function</i>; Inverse of a function
• Online Access: http://www.iopan.gda.pl/oceanologia/44_2.html#A1

The paper deals with the conformal mapping of a plane, finite, simply connected domain, representing the southern segment of the Gulf of Gdansk, enclosed from the North by a parallel, tangential to Cape Rozewie. The segment contains the Hel Peninsula. <br> The method of <i>double</i> decomposition, presented in Prosnak &amp; Czesnik (2001), is applied to the transformation of such an <i>original</i> domain onto a <i>canonical</i> one, which consists of two separate unit discs. <br> The <i>first</i> decomposition concerns the domain which is split into two adjacent subdomains by means of a segment of a straight line. <br> The <i>second</i> decomposition involves two holomorphic functions, each one mapping a subdomain onto a separate disc. The decomposition consists in replacing the function by a sequence of simple ones, so that the mapping is performed step-wise. Each sequence starts with the <i>Schwarz-Christoffel</i> function, the last step consisting in an inversion that transforms an infinite circular domain onto a disc. <br> <i>The data</i> for the problem is contained in the Annex, and is represented by two sets of rectangular coordinates defining consecutive discrete points of the contours bounding the subdomains. <br> <i>The solution</i> to the problem consists of: <br> - two sets of functions, consecutively transforming each of the subdomains; <br> - the numerical values of the parameters of these functions; <br> - a set of figures illustrating the consecutive transformations. <br> The accuracy of the first, and of the penultimate transformation are given, because only in the case of these two functions do the unknown coefficients have to be determined by means of a suitable iterative process. The coefficients of all the remaining functions are evaluated from exact formulae. <br> It should be recalled that the depth of the Gulf of Gdansk varies considerably - from a few to 110 metres - the gradients of the bottom being rather large. Therefore, the domain of the solution for any mathematical problem describing the hydrodynamical phenomena occurring in the Gulf is usually taken to be <i>three-dimensional</i>. Nevertheless, the paper deals with the transformation of the <i>free surface</i> of the Gulf, <i>assumed as plane</i>. It should be emphasized that this assumption does not mean that the whole domain of the solution has to be regarded as plane.