Title
Analytical solution to the 3D tidal flow with vertically varying eddy viscosity
Members’ names and affiliations
NameInstitution and Department Employment position E-mailWensheng Jiang

Ocean University of China

Professor

wsjang@ouc.edu.cn

Yang Chen

Ocean University of China

Ph. D. Student,
Grade 3chenyang@stu.ouc.edu.cn

Aim
The tide is often the dominant process in shallow seas and bays; it plays an importantrole in water mixing and mass transport. Therefore it is essential to understand the tidal
motion accurately, especially the spatial characteristics, which is meaningful tooceanography and ocean engineering. The aim of this study is to get the analyticalsolution to the 3D tidal flow with vertical varying eddy viscosity to improve the studiesof the inter-tidal residual circulation in the coastal sea. This can be a basis for further
study of the movement of the environment-related substances in the costal water. Anotheraim of this project is to enhance the cooperation between the College of Environmental
Science and Engineering of OUC and the Center for Marine Environmental Studies ofEhime University.
Procedure
The deduction of the analytical solution of the 3D tidal current with verticallyvarying eddy viscosity in a narrow bay is examined by Prof. Wensheng Jiang and Prof.
Xinyu Guo. Its application in the study of the tide-induced residual current is also
discussed. During the visit a presentation “The Lagrangian residual current and its
application in analytical, numerical and experimental studies in a narrow bay” wasgiven by Prof. Wensheng Jiang in CMES, Ehime University. Prof. Jiang also visited thelaboratory of the Center for Marine Environmental Studies accompanied by Prof. Xinyu
Guo. They discussed the further cooperation between the two groups.
Results
The 3D tidal current is solved in a semi-enclosed rectangular bay in this study. Thex and y coordinates are along the two horizontal sides of the bay with x = 0 being at
the open boundary and x = L being at the head of the bay. The lateral boundaries arey = yb and y = ye . The surface of the still water is set at z = 0 and z = −h is the sea

1 / 4bottom. The tidal signal is imposed at the open boundary so the tidal current is the main
movement in the area.
Assume the variables to have the following characteristic value: xc =  , yc = B ,zc = hc and tc = 2 / c , where  = ghc / c is the wave length, hc is the typical water

depth of the sea area and c is the circular frequency of tide,  c is the characteristic
value of tidal elevation.
To solve the tidal equations, the double-perturbation is used to expand the tidal
equations with regarding to small parameters κ and δ2 , where  = B /  ,  =  c / hcand  =(  / /h ) with  being the characteristic value of the vertical eddy
2ccccviscosity. The zeroth order tidal current are listed below.
  u0 = 0(1)

u0

u

= − 0 +  ( 0 )txzz0=−(2)

 0
y(3)

At the sea surface, z = 0 , 0t (u0 , v0 )=0z
w0 =(4)At the sea bottom, z = −h ,u0 = 0(5)

At the fixed boundary,0

 v dz = 0 at y = y and y−h 0be0

 u0dz = 0 at x = L(6)−hAt the open boundary, x = 0 , 0 =  open
According to the previous study the vertical eddy viscosity is often of parabolic
form, so the expression of the vertical eddy viscosity is assumed as ( z ) = a1 ( z + a2 )( z + a3 ) in this study. As seen in Fig. 1,  has the minimum value
2 / 4(7)R (0  R  1) at z = − h0 and attains its maximum value 1 at z = zm = − h0 hm , whereh0 is the maximum value of the water depth.

Solutions are presented in a bay with the water depth varying along the transverse
direction:cos( ( L − x))cos( L)i sin(  ( L − x))
U0 =qcos(  L)
N0 =

V0 = −(8)(9)

i  2G cos(  ( L − x))q
P0 cos(  L)

(10)zi  2 cos(  ( L − x)) −2 G dP0 z
W0 =[−( + 2 )− h qdz + G − h qdz]cos(  L)
P0 P0 dyP0y(11)

where N 0 , U 0 , V0 and W0 are amplitude of the water elevation, velocities in thethree directions, respectively. The parameters in the above formula are in the followingforms:
1y000−h −2 = −  P0 dy is a constant, G = − P0 ( y)dy − y −2 and P0 =  qdz . q is thesolution of a non-homogeneous hypergeometric equation. The results are drawn in Fig. 2.

Fig. 1 The profile of the non-dimensional vertical eddy viscosity.

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Fig. 2 Velocities at six tidal phases for three different values of  ( R = 0.15 , hm =1/4 ). The
section is located at mid-bay. The leftmost column corresponds to the phase of high water. The axialvelocity is negative in the shaded area; lateral and vertical velocities are represented by arrows.

Publication/conference presentation
It is under preparation.
Perspectives in future
The analytical solution of the 3D current with varying eddy viscosity in a narrow bay
can be used as a benchmark for the numerical model. It is also a basis for the further
acquisition of the tide-induced residual current in the same model bay. It can enhance
the understanding of the residual current in a real bay.

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