Subsurface Flow Generated by a Steady Wind Stress Applied at the
Water Surface
by
Lionel Gurfinkiel
A Thesis Submitted to the Faculty of
The College of Engineering
In Partial Fulfillment of the Requirements for the Degree of
Master of Science
Florida Atlantic University
Boca Raton, Florida
August 2003 SUBSURFACE FLOW GENERATED BY A STEADY WIND STRESS
APPLIED AT THE WATER SURFACE
by
Lionel Gurfinkiel
This thesis was prepared under the direction of the candidate's thesis advisor, Dr. Manhar Dhanak, Department of Ocean Engineering, and has been approved by the members of his supervisory committee. It was submitted to the faculty of the College of Engineering and was accepted in partial fulfillment of the requirements for the degree of Master of Science.
SUPER VISOR Y COMMITTEE:
Dr. Manhar Dhanak
Dr. P. Ananthakrishnan 0.'1:-~ Dr. Oleg Zikanov
Chairman, Department of Ocean Engineering
Date
II ACKNOWLEDGEMENTS
The author would like to extend his utmost thanks and gratitude to Dr. Manhar
Dhanak, thesis advisor, for his precious advice in performing this research and for his
patience and willingness to take the time to guide and assist in the work involved in
completing this thesis. Gratitude is extended to the supervisory committee, Dr. P.
Ananthakrishnan for his valuable input throughout this project, and Dr. Oleg Zikanov
for his indispensable help in the computer code work and his expertise in this project.
A special thanks is given to Dr. Pierre-Emmanuel Guillerm for his patience and his
expert advice in the area of computational fluid dynamics. Special thanks are also
given to the faculty and staff of the Department of Ocean Engineering at Florida
Atlantic University for the opportunity and support in accomplishing this research.
This work was supported by the Office of Naval Research under grant N00014-00-l-
0218 (Program Managers: Dr. Thomas Curtin and Dr. Thomas Swean).
1Il ABSTRACT
Author: Lionel Gurfinkiel
Title: Subsurface Flow Generated by a Steady Wind Stress Applied at the Water Surface
Institution: Florida Atlantic University
Dissertation Advisor: Dr. Manhar Dhanak
Degree: Master of Science
Year: 2003
A turbulent water current induced by winds, through a friction force at the sea surface
and subjected to the Coriolis force in shallow water was studied. A Large Eddy
Simulation model developed by Zikanov et al. is used to solve the Navier-Stokes
equations. To define the bottom boundary condition, a drag coefficient parameter,
based on the ideas of Csanady, is used to evaluate the shear stress at the bottom. To find a suitable bottom boundary condition for this LES simulation, several cases were considered with change in drag coefficient property. The effect of variation in the depth of the water column was also considered. Variation in surface deflection of the current, variation of the mass flux and distribution of eddy viscosity with depth of the water column are determined. The cases are compared with the case of a deep water column. Numerical results are also compared with field observations.
IV TABLE OF CONTENTS
TABLE OF FIGURES ...... vii
1 INTRODUCTION ...... l
1.1 Objective ...... 1
1.2 Background ...... 2
1.2.1 Ekman solution ...... 3
1.2.2 Ekman transport ...... 7
1.2.3 Limitations of the Ekman solution ...... 9
1.3 Relationship with the thesis research ...... 10
2 LARGE EDDY SIMULATION METHOD ...... ll
2.1 Comparisons with the Direct Numerical Simulation ...... 11
2.2 Presentation of the LES model ...... 12
3 APPROACH ...... 16
3.1 Assumptions ...... 16
3.2 Coriolis force ...... 17
3.3 Governing equations ...... 18
3.4 Boundary conditions applied in deep water ...... 19
3.5 Boundary conditions applied in shallow water...... 20
4 COMPUTATIONAL METHOD ...... 25 v 4.1 LES model used ...... 25
4.2 Numerical method ...... 27
4.3 Domain of the simulations ...... 28
5 RESULTS ...... 31
6 DISCUSSION ...... 36
6.1 Original simulated flow in shallow water compared with deep water
simulation ...... 36
6.2 Variation of drag coefficient simulations ...... 46
6.3 Variation of the depth ...... 50
6.4 Ekman transport ...... 55
6.4.1 Mass transport in deep water ...... 55
6.4.2 Mass transport in shallow water ...... 57
6.5 Comparison with the field observations ...... 58
7 CONCLUSIONS AND FUTURE WORK ...... 64
APPENDIX ...... 67
REFERENCE ...... 69
VI TABLES OF FIGURES
Figure 1-1 Hodograph showing velocity at various depths ...... 6
Figure 1-2 Perspective view showing velocity decreasing and rotating with increase
of depth of an Ekman flow ...... 8
Figure 2-1. Schematic representation of turbulent motion (left) and the time
dependence of a velocity component at a point (right) [7] ...... 12
Figure 2-2: Filter G(r) : box filter, dashed line; Gaussian filter, solid line; Sharp
spectral filter, dot-dashed line ...... 13
Figure 5-1 Horizontally and time-averaged profiles of the mean horizontal current u
and v ...... 32
Figure 5-2 Velocity hodograph ...... 33
Figure 5-3: Subgrid grade and Reynolds stresses ...... 33
Figure 5-4: effective viscosity coefficient ...... 35
Figure 6-1: volume-averaged kinetic energy as a function of time for shallow water
flow simulated at latitude 26 oand wind direction being 170° from south-north
direction ...... 37
Vll Figure 6-2 volume-averaged kinetic energy as a function of time for deep water flow
simulated at latitude 26 oand wind direction being 170° from south-north
direction ...... 38
Figure 6-3: Horizontally and time-averaged profiles of the components of the mean
horizontal current u and v of shallow water at latitude 26 oand wind direction
being 170° from south-north direction ...... 39
Figure 6-4 Horizontally and time-averaged profiles of the components of the mean
horizontal current u and v of shallow water at latitude 26 oand wind direction
being 170° from south-north direction ...... 40
Figure 6-5 Speed of current of the simulated flow for shallow water at latitude 26°
and wind direction being 170° from south-north direction ...... 41
Figure 6-6 Speed of current of the simulated flow for deep water at latitude 26° and
wind direction being 170° from south-north direction ...... 41
Figure 6-7 Effective viscosity coefficient of the simulated flow for shallow water at
latitude 26° and wind direction being 170° from south-north direction ...... 42
Figure 6-8 Effective viscosity coefficient of the simulated flow for deep water at
latitude 26° and wind direction being 170° from south-north direction ...... 43
Figure 6-9 Root-mean-square velocity fluctuations of the simulated flow for shallow
water at latitude 26° and wind direction being 170° from south-north direction 44
Figure 6-10 Root-mean-square velocity fluctuations of the simulated flow for deep
water at latitude 26° and wind direction being 170° from south-north direction 44
Figure 6-11 Velocity hodograph of the simulated flow for shallow water and deep
water at latitude 26° and wind direction being 170° from south-north direction.45
V111 Figure 6-12 Variation of Drag Coefficients with the time for di fferent values of the
constant ci ...... 47
Figure 6-13 volume-averaged kinetic energy as a function of time for shallow water
flow simulated at latitude 26 oand wind direction being 170° from south-north
direction with ci = 5.35 (on the left) and ci = 21.4(on the right). ·· ··· ····· ······ ···· 48
Figure 6-14 Velocity hodograph of the simulated flow for shallow water with changes
of the drag coefficient properties ...... 49
Figure 6-15 Horizontally and time-averaged profiles of the components of the mean
horizontal current u and v of shallow water simulation with a depth of 0.1 , 0.2
and 0.3. Dotted line represent u and v of the deep water simulation ...... 51
Figure 6-16 Speed of current of the simulated flow for shallow water simulation with
a depth of 0.1 , 0.2 and 0.3. Dotted line represents the speed current of the deep
water simulation ...... 53
Figure 6-17 Velocity hodograph of the simulated flows for shallow water with
changes of the depth and simulated flow in deep water...... 54
Figure 6-18 Perspective view showing velocity decreasing and rotating with increase
of depth of deep water flow simulated at latitude 26° with a wind stress having
an angle of 170° with the South-North direction ...... 57
Figure 6-19 Perspective view showing velocity decreasing and rotating with increase
of depth of shallow water flow simulated at latitude 26° with a wind stress
having an angle of 170° with the South-North direction ...... 58
IX Figure 6-20 Wind vectors from meteorological station displaying change in wind
strength and direction as low-pressure cold front passes through region (M.
Chernys Thesis [17]) ...... 59
Figure 6-21 Ocean current contours as recorded by the bottom-mounted ADCP
measured in meter/seconde. Top panel represents North component and middle
panel represents East component of velocity. Bottom panel shows the depth
averaged North (solid line) and East (dashed line) components over the
experiment period (M. Chernys Thesis [17]) ...... 60
Figure 6-22 Perspective view showing velocity decreasing and rotating with increase
of depth of the observed flow ...... 61
Figure 6-23 Velocity hodograph of the simulated flow for shallow water and of the
observed flow ...... 62
X 1 INTRODUCTION
1. 1 Objective
The objective of the research is to simulate a flow created by a steady wind stress applied at the water surface with physical parameters corresponding to South Florida waters. It has been shown that the turbulent flow generated depends on conditions such as the latitude of the site and the direction of the wind [1]. We aim to simulate the flow in shallow water for specific latitude of 26° N for winds blowing from different directions, analyze the simulated data and compare with field observations.
The computer code used in this research is the one developed by Zikanov et al. [1] for deep water application. For the shallow water simulation, the code needed to be modified to allow for appropriate boundary conditions at the bottom. This thesis therefore represents an extension to results of Zikanov et al.
The numerical simulation obtained with local parameters will be compared with observed field data obtained from experiments in littoral water offshore of South
Florida.
1 1.2 Background
Most of the surface currents of the ocean are in geostrophic balance. The first
significant advance in the study of the wind-driven ocean circulation was made by
Fridtjof Nansen and Vagn Walfrid Ekman in the early part of the twentieth century.
Nansen, a biologist, led an expedition for the scientific exploration of the Arctic
Ocean. By locking his ship in a big ice pack, many observations were made. They
included the winds (speed and direction) and the position of the ship (and therefore
the ice pack). His observations revealed that the ship and thus the ice pack, did not
drift directly downwind; rather it moved consistently some 20° to 40° to the right of
the wind.
Ekman [2] worked out a mathematical theory of current induced by wind acting as a
friction force at sea surface subjected to the Coriolis force.
In 1995 T.K. Chereskin [16] studied wind driven currents observed in the Californian
seas for comparison with the Ekman theory. To do that, measurements were made
over a 6-month period with a moored acoustic Doppler current profiler (ADCP) and a
wind sensor mounted at the top of a buoy. A 4-month period of steady unidirectional winds and accompanying currents was observed and the observations were used to compare with the Ekman flow. The wind-driven flow was separated from pressure driven flow by subtraction of a deep reference current. The wind-driven flow was shown to be in Ekman balance on timescales as short as a few days. A net transport to
2 the right of the wind was also shown. The mean velocity observed was a smooth
spiral flatter than the theoretical Ekman spiral.
1.2.1 Ekman solution
In this section, a review of the Ekman [2] solution for a wind driven current is
presented.
A turbulent Ekman layer is created in the water column by the presence of a steady wind at the surface. The generation of currents by wind requires frictional transfer of momentum from the atmosphere to the ocean. The effect of wind stress on oceanic movement is best analyzed by excluding for the moment the effect of the pressure field on the balance of forces and assuming that the ocean is unstratified, that is, of uniform density, and its surface horizontal. Ekman layer is the result of a balance between friction and the Coriolis force. Since wind blowing over water is always associated with turbulent mixing, the condition of uniform density is usually satisfied in the wind affected surface layer, and the balance between friction and the Coriolis force prevails. This layer is often called the Ekman layer. In a non-rotating frame of reference this would result in water movement in the direction of the wind. Rotation gives rise to an apparent force (the Coriolis force) which acts perpendicular to the direction of movement. According to the Ekman solution, the combined action of friction and the Coriolis force produces a surface current directed at 45° to the wind direction and further deflection from the wind direction down the water column.
3 Following Ekman, assuming steady state conditions, a homogeneous ocean, and uniform blowing wind, if wind stress and internal frictional stresses (Reynolds stresses) exactly balance the Coriolis force, the horizontal equations of motion are simplified to:
(1-1)
(1-2) where f = 2.QsinA is the Coriolis parameter with .Q the Earth's rotation and A the latitude and Az is the vertical eddy viscosity coefficient, and u£ and v£ are the current velocity components in the x direction (east) and y direction (north), respectively, with the z direction positive upwards.
The surface boundary condition is the specification of the wind stress at the surface
(z=O). To simplify, only a non-zero x-component to the surface wind stress rx is assumed. Boundary condition at the surface becomes
du rx =pAz-, at z = 0 and (1-3) dz
0 = dv at z = 0. (1-4) dz'
At great depth z = -oo, the velocity is assumed to tend to 0. So we have the following boundary conditions
u = v = 0 , at z = -oo (1-5)
4 Solving these equations, subject to the boundary conditions and assuming A2 is constant, we obtain, relative to the wind direction, the Ekman velocity profile (see the appendix for details)
(1-6) in northern hemisphere.
r DE t D . z z v E =-x-ez E(sm--cos-) (1-7) 2pA2 DE DE where rx =magnitude of the wind stress on the sea surface. DE is the Ekman depth,
(1-8) f is the Corio lis parameter, f = 2.Q sin¢, .Q is the rate of Earth's rotation and ¢ is the latitude. DE is known to be equal to about a few tens of meter in the ocean.
In term of the current speed V = .J u 2 + v 2 and the current direction given by angle B, with respect to the wind direction, measured in the clockwise direction we have
(1-9) and
. z z sm - -cos- DE DE tanB =- (1-10) . z z sm - +cos- DE DE
5 Equation (1-9) shows that the speed of the horizontal current decreases exponentially with depth, with the Ekman depth as the scaling depth.
It also shows that the speed of the surface current increases with increasing wind stress, decreasing turbulent viscosity and decreasing distance from the equator.
Equation (1-10) shows that at z = 0 we obtain 8 = 45° , at z = -nD E I 4 we obtain
8=90° andat z =-JrDE/2 weobtain 8=135° .
That means that the surface current flows at 45° to the right (left) of the wind direction and spirals further to the right (left) with depth in the northern (southern) hemisphere. The direction of the flow becomes opposite to that at the surface at z=-DE.
.. t
ff --...... ;::~. 8 -r/6.:0 .:--- ~&'~
Figure 1-1 Hodograph showing velocity at various depths.
6 To describe the circulation in the real ocean the presence of the three major forces, the pressure gradient, the Coriolis force and the friction should be considered. This was first achieved by the oceanographer Hans Ulrik Sverdrup [17], and the balance of forces found in most of the ocean, is therefore known as the Sverdrup balance. This says that friction is mainly important in the Ekman layer at the surface, below which all movement is approximately friction free. Movement below the Ekman layer is induced by transfer of water from the Ekman layer to the depths below as a result of flow convergence or divergence in the Ekman layer. In the case of flow convergence, water is forced downwards to the depths below; this process is known as Ekman pumping. In the case of flow divergence, water is pumped upwards into the Ekman layer; this process is known as Ekman suction (or negative Ekman pumping).
1.2.2 Ekman transport
The Ekman transport is the average flow in the Ekman Layer in a water column down to one or two Ekman depths. Because the flow decreases with depth, there must be very little friction at the bottom of the Ekman layer. For the Ekman layer as a whole, there is a balance between wind friction and the Coriolis force. This mean that the average flow in the Ekman layer must be perpendicular to the wind: to the right in the
Northern Hemisphere and to the left in the Southern Hemisphere (see figure 1-2).
7 Figure 1-2 Perspective view showing velocity decreasing and rotating with increase of depth of an Ekman flow.
In term of stresses we have:
0 = pfv+ dTxz dz
dryz 0=-pju+-- dz
Integrating these equations vertically to some depth Z yields:
o o dr 0=-J fpudz + f a: z dz =-fMx +Tyz (O)+Tyz (-Z) -z -z with M x and M Y being the vertically-integrated mass transports m the x and the y directions:
8 0 M x = fpudz -z
M and M have units of mass/length/time. M Y, for example, is the mass of water X y
down to depth Z flowing north per unit time per unit east-west distance. Assuming Z
is large enough so that the current are weak and friction small, so we have
1:x M =-- Y f
r r x y where X and y are the wind stress components in the and directions.
1.2.3 Limitations of the Ekman solution
The above theory is well known and clearly observed in the laboratory where the viscosity is molecular and constant but has probably never been observed in the open ocean.
Ekman made many assumptions to his model. He assumed that no boundary was considered. Even if this is not realistic, this assumption is not too bad if we consider a flow away from the coast. The flow is in infinitely deep water. If we consider the
Ekman depth DE with an order of 100-200 m compared with the averaged ocean depth of 4000m, this source of error in the open ocean is small.
9 The main limitations of the Ekman theory are the assumptions of a constant effective
viscosity coefficient A2 and the absence of consideration of the tangential component
of the Earth's rotation vector. The assumption of a constant effective viscosity A2 allows analytic solution of the problem, but is believed not to be correct.
1.3 Relationship with the thesis research
Zikanov et al.(2002) revisited the Ekman theory. They show that Ekman did not represent adequately an actual flow near the ocean surface. They investigate the consequences of the variability of the turbulent viscosity coefficient and the flow dependence on the latitude and wind direction. A numerical model with a Large Eddy
Simulations was developed to simulate a turbulent Ekman layer generated by wind blowing uniform] y at the water surface. Here we utilize Zikanov et al.' s computer code to study the effect of inclusion of bottom boundary in the problem.
10 2 LARGE EDDY SIMULATION METHOD
The Large eddy Simulation (LES) is the method used by Zikanov et al.(2002) to
evaluate the full equation of motion and to show the weakness of the Ekman solution
for a flow generated by a wind blowing uniformly at the sea surface. This method is
less accurate but much less time consuming than Direct Numerical Simulation.
2. 1 Comparisons with the Direct Numerical Simulation
The Direct Numerical Simulation (DNS) is currently the most accurate numerical
approach of turbulence simulation. It solves the governing equations through
numerical discretizations. All motions contained in the flow are resolved. For
accurate simulation, the computational domain must be at least as large as the largest
turbulent eddies of the flow and the grid must be as small as the smallest eddies.
Because of these conditions and of the limitations of the computer speed and
memory, the DNS can be successful for flow of relatively low Reynolds number.
Turbulent flows contain wide ranges of length and time scales. Large motions are generally much more energetic than small ones.
Large-Eddy Simulation (LES) extends the utility of DNS for practical applications by intentionally leaving the smallest turbulent structures spatially under-resolved (see
11 figure 2-1). These are then represented by a subgrid stress model which accounts for
contributions to the flow field which are not resolved by the numerical algorithm on
the computational mesh. Because a coarser grid is employed than that required for
DNS, a substantial saving in computer resources is realized.
LES DNS u
--DNS ·• ··•·• •· LES
Figure 2-1. Schematic representation of turbulent motion (left) and the time dependence of a velocity component at a point (right) [7].
2.2 Presentation of the LES model
Only large scale features of the velocity field are considered. This is best done by filtering (Leonard,1974 [11]). The large or resolved scale field (the one to be simulated) is a local average of the complete field.
The filtered velocity is defined by:
(2-1) where the filter function G (figure 2-2) is a localized function. The filter functions include a Gaussian and box filter and a cutoff. A length scale ~is associated with the
12 filter. In a rough sense, eddies of size larger than !1 are large eddies and will be
simulated while those smaller than !1 are small eddies and will be modeled.
tO I -~ . ,*'f ~
: ' j I I ,, " f I I I \ I' I :1 '• , I I j' 1' I )
I I • .. I ' ' '. ' . ' ..; ..... - ~ ' I ' /I ' ' ' - ' '- I ' I ' \ -. \ , ,, ' j ' . '
-2 0 2 4 riA
Figure 2-2: Filter G(r) : box filter, dashed line; Gaussian filter, solid line; Sharp spectral filter, dot-dashed line.
The filtered Navier-Stokes equations are equal to
(2-2) and the filtered continuity equation is given by
(2-3)
13 Because wur:f-u iu 1 and because the quantity on the left side of this inequality is not
easily computed, a modeled approximation for the difference between the two sides
of this inequality is introduced. It is called the subgrid scale Reynolds stress.
(2-4)
The models used to approximate the SGS Reynolds stress are called subgrid scale
models. the most commonly used is an eddy viscosity model proposed by
Smargorinsky (1963) [3]. The principal effects of the SGS Reynolds stress are
increased transport and dissipation. These phenomena are due to the viscosity in
laminar flows. The following model was used by Smargorinsky
(2-5)
1 where f.l1 is the eddy viscosity and Su=1 Caa~ i + aau ) is the strain rate of the large scale 2 Xi Xi or resolved field. The eddy viscosity is modeled as j.JJ=C} p~2 jsj where Cs is a model
2 parameter to be determined, ~ is the filter length scale and I~=(&J.~;J }' • Lilly[12] determined that Cs ~0.23 for isotropic turbulence with cutoff in the inertial subrange and ~ equal to the grid size. However, for shear-driven turbulence, Deardorff [13] used a smaller coefficient Cs=O.l. But a single universal constant was not possible to establish. In 1990 Germano et al.[4] developed a Dynamic subgrid scale eddy viscosity model. It removes many of the difficulties:
14 In the shear flows, the Smargorinsky model parameter needs to be much smaller than in isotropic turbulence. The dynamic model produces this change automatically.
The model parameter needs to be reduced even further near walls. The dynamic model automatically decreases the parameter in the correct manner near the wall.
The definition of the length scale for anisotropic grids or filters is unclear.
This issue becomes moot with the dynamic model because the model compensates for any error in the length scale by changing the value of the parameter.
15 3 APPROACH
In this chapter, we will present the numerical model used in this thesis for computing
the flow. We consider a turbulent flow created by a steady wind near the ocean
surface, taking account of the influence of the earth's rotation.
One of the objectives of this research is to run the flow simulation with parameters
pertaining to conditions in shallow waters off South Florida. The simulation in deep
water has a free-slip bottom condition. The same simulation could be done with a
smaller depth and with a non-slip bottom. The bottom boundary conditions need to be
changed. Zikanov et al. 's [1] Fortran code has been modified to run cases
corresponding to flows generated in shallow water. Some simulations with these new
elements have been computed and analyzed. The boundary conditions at the bottom
and the grid size were changed.
3. 1 Assumptions
These assumptions are considered:
The fluid is incompressible and neutrally stratified.
The flow is statistically homogeneous in the horizontal directions. So the
effect of Ekman pumping is ignored
16 The water surface is approximated by an impermeable unmoving flat lid. The
effects of surface gravity waves are neglected.
The typical Reynolds number of a real ocean flow is very high, so we do not
resolve the viscous boundary layer at the upper surface. Instead, the boundary
condition is prescribed in the form of a constant shear stress. The molecular
viscosity term is neglected in the equations. Only the momentum transfer
produced by the turbulent stresses is considered.
3.2 Coriolis force
Effect of Coriolis force are included in this model. Coriolis showed that, if the ordinary Newtonian laws of motion of bodies are to be used in a rotating frame of reference, an inertial force must be included in the equations of motion.
The effect of the Coriolis force is an apparent deflection of the path of an object that moves within a rotating coordinate system. The object does not actually deviate from its path, but it appears to do so because of the motion of the coordinate system.
On the Earth an object that moves along a north-south path, or longitudinal line, will have an apparent deflection to the right in the Northern Hemisphere and to the left in the Southern Hemisphere. The two reasons for this phenomenon are the Earth rotation eastward and the tangential velocity of a point on the Earth being a function of latitude (the velocity is essentially zero at the poles and is maximum at the Equator).
17 3.3 Governing equations
The governing equations are based on the law of conservation of momentum and on the law of conservation of mass.
The non dimensional governing equations are:
Navier-Stokes Equations
au - ap arlk 'J 0 -at + u · Vu = --ax + --axk + v - w cot /L sm y, (3-1)
av - ap ar2k 'J -at + u · Vv = --ay + --axk + u - w cot /L cosy, (3-2)
aw - ap ar3k 'J 'J 0 -at + U · Vw = --az + --axk + VCOt/LCOS y- UCOt/LSln y, (3-3)
Continuity Equation
(3-4)
Here A is the latitude, y is the angle between the wind direction and the south-north direction. The }-plane' approximation is not used since the full Coriolis force is considered. The Coriolis deflection is related to the motion of the object, the motion of the Earth, and the latitude. The non-dimensional components of the Coriolis force associated with the tangential component Qr of the Earth rotation vector are given by
1 Q n: = (-)cotAcosy, and 2
Q 1)! = (.!_)cotA-sin y 2
18 In this model, the equations are non dimensionalized. The typical velocity scale is the
surface friction velocity u. = with r a wind stress constant applied at the vP:~, 0
surface and p 0 is the constant fluid density. The time scale is the reciprocal of the
1 Coriolis parameter, 11 f = with Q the angular velocity of the earth 2Qsinlt'
rotation, is used.
The length scale is the turbulent length L = u. If . The scale for pressure p and
turbulent stresses t";k is the surface shear stress r 0 = p 0u?
3.4 Boundary conditions applied in deep water
The domain of the model is rectangular with the dimensions Lx x LY x Lz .
In the horizontal directions, the boundary conditions are periodic. For the deep water
simulation, at the bottom of the domain, a free slip condition is imposed when no
boundaries are present:
w = r 13 = r 23 = 0 at z = 0
The upper surface is assumed to be an impermeable lid with a constant shear stress
w=O, r 13 = g(t)cos[a(t)], 1"23 = g(t)sin[a(t)], at z=Lz
with g(t) a non-dimensional amplitude and a(t) the direction of the surface wind
stress.
Zikanov et a1.[1] assumed a deep ocean and imposed a free-slip condition at the bottom in the numerical model. Data obtained from offshore experiments in the 19 Atlantic Ocean of South Florida were taken in shallow water. To compare these data with numeric simulation, an extension of Zikanov et al. is implemented here by modifying the bottom boundary conditions. A rigid bottom will be imposed and the bottom boundary conditions are changed as describe below.
3.5 Boundary conditions applied in shallow water
In shallow water, the free-slip condition at the bottom of the computational domain does not hold. However, it is also not right to set the velocity components equal to zero at the bottoms since, in view of the grid spacing of the Large Eddy Simulation method, details of the bottom boundary layer are not captured. The grid space in the vertical direction at the bottom must be very small to consider accurately the effects of the boundary layer. For this reason, at the bottom boundary, the velocity is not set to zero but the components of the bottom stress are specified, and given by the relationship involving the drag coefficient for bottom friction,
The drag coefficient used m the simulation is based on consideration of G. T.
Csanady (1967, [14]).
Csanady used similarity hypotheses analogous to those used in two-dimensional turbulent boundary layer theory to establish a functional form of a resistance law, connecting geostrophic drag coefficient to surface Rossby number which is the ratio of the nonlinear terms in the equation of motion to the Coriolis term. This was done 20 in much the same way as for the corresponding law established for two-dimensional
boundary layers.
In the Ekman layers, Csanady considered what determines the depth of the layer in turbulent flow. For high Reynolds numbers and in the absence of density- stratification, the only independent variables are a typical velocity U , the Coriolis
parameter f and the length z0 characterizing the roughness of the surface. The velocity U is the magnitude of the free stream velocity if the surface is fixed and the magnitude of the vector difference between the free-stream velocity and the surface velocity otherwise.
The depth of the layer D, the friction velocity on the surface u * = ~r0 I p, and the direction of the surface shear stress a are functions of the only three variables, f,
U and z0 , given by
D = fA (f, U, Z0 )
u • = f/J2 (f, U, Z0 ) (3-5)
a = f/13 (f, U, Z0 )
In non-dimensional form these quantities can be written as
(3-6)
u • I U is the square root of the drag coefficient C d , so we have
21 (3-7)
U I fD is the Rossby number and U I Jz 0 is the surface Rossby number. By
eliminating the surface Rossby number from the first two equations of equations (3-6)
Csanady obtained
(3-8)
(3-9)
He demonstrated that the layer depth D IS related to the drag coefficient at the
surface.
Csanady considered the Ekman layer to be similar to a two-dimensional turbulent
boundary layer. The two-dimensional layer (see Townsend in 1956 [19], for example)
is composed of a "wall layer" where the velocity distribution is determined by wall
shear and roughness lengths (or fluid viscosity for a smooth surface), and an "outer"
layer where the "velocity defect" profile is determined by wall shear and layer
thickness. Between the two layers, there is a region of overlap, where the velocity
distribution is logarithmic.
Csanady used complex velocity vectors to postulate the existence of a "law of the wall' near the boundary surface in a turbulent Ekman layer and a "velocity defect law". Assuming an overlap region of the two layers, the following formulas were found through matching,
U u * -cosa= 2 5ln-+B-C (3-10) u * . fzo r
22 -u. s1na=-. c ; (3-11) u
with a being the angle between r 0 and the velocity-defect vector at the boundary having the magnitude equal to U . This value has been used as a convenient velocity
scale in the definition of the drag coefficient in Equation (3-7). The first equation of
(3-6) can be written as
sin a = -C; .JC: (3-12)
u· = 2.5ln-+B-C (3-13) F fzo r Csanady used the characteristics of the complete Ekman layers analyzed by Lettau
and his collaborators (Lettau, 1962 [20]; Lettau and Roeber, 1964; Johnson, 1962
[21]) to determine the constant C; to be equal to 10.7 giving the following equation
sin a= -10.7/C: (3-14)
He also found that B- C, = 3.8 which gave the following equation for the drag coeffi ci en t:
1 c =------~=------(3-15) d u.JC: (2.5xln( d )+3.8)2 +10.7 2 fzo
This Drag coefficient definition is used in the program to determine the Bottom stress necessary for the bottom boundary conditions. Using the equation (3-7), the non- dimensional components of the bottom stress are
r bx = CdU(ucos(a)- vsin(a)) (3-16)
r by = CdU(vcos(a)+usin(a)) (3-17)
23 Because the drag coefficient value Cd is in both sides of the equation (3-15), an arbitrary value of cd is first given to the program, then the program find the converging value of Cd by making a loop on the equation (3-15).
In the program the roughness of the bottom z0 is replaced by the vertical grid spacing
L1z at the bottom.
U is taken to be the current magnitude near the surface of the flow
24 4 COMPUTATIONAL METHOD
4. 1 LES model used
In this model, following Zikanov et al., the dynamic subgrid-scale closure is used
(Germano et al. 1991[4] and Lilly 1992[5]). To separate the small eddies from the
large ones, two filtering operators are used. The first one is the grid filter G , denoted
by an overbar:
]ex)= jGCx,x')f(x')dx' (4-1)
The integral is extended to the entire computational domain.
Equations of motions including the grid filter are
au-·+ a (u-:u- ·)= a p- a ""+ 1 azu; ""':\" I ~ I } ~ -:::1-. ~I} D :::1 .:::1 . ' (4-2) at UXJ OX1 UXJ H e UXJUXJ
(4-3)
Where iii and p represent filtered fields. The effects of the small scales appear in the subgrid-scale stress term
(4-4) which is modeled. The second, coarser spatial filter is called the "test" filter G , denoted by a tilde:
f(x)= fccx,x')f(x')dx' (4-5)
25 the filter width of the test filter is assumed to be larger than that of the grid filter.
We have G=GG
The filtered Navier-Stokes equations become:
(4-6)
The new subgrid-scale stress is
(4-7)
the resolved turbulent stress is defined as
(4-8)
The S closure, applied to the SGS stress is given by
(4-9) with &j =1 when i= j and zero otherwise, sij =1 cii;;;i +au j) is the strain rate tensor, 2 dXj ()xi
1~=(2.5;j ,5;j ) 112 is the magnitude of large-scale strain rate tensor, ~=(~ x ~ y ~ z ) 1 13 is the filter width proportional to the grid size, C is the dynamically adjusted Smargorinsky coefficient.
The subtest-scale stress Tij is similarly approximated by
(4-10)
26 A proper choice of C should be made for a consistency between Equations (4-3) and
(4-7). This is shown by subtraction of the test scale average of TiJ from TiJ to obtain
L J=Iij -Tij =-UiUJ+U;U I (4-11)
Elements of L are the resolved components of the turbulent stress tensor associated
with scales of motion between the test scale and the grid scale. These scales can be
evaluated and compared locally with the S closure approximation by subtracting the
test scale average of (4-3) and (4-7)
(4-12)
Where (4-13)
By applying a least squares approach, Lilly (1992) defined the square of the error in
(4-11) to be
(4-14)
the minimum of Q is given by
C 1 LJM iJ (4-15) 2 MJ
4.2 Numerical method
For the numerical simulation, Zikanov et al. computer code uses a combination of pseudo-spectral and finite-difference methods to calculate the flow.
In the horizontal directions, the pseudo spectral technique based on the Fast Fourier
Transform is used since the principle of periodicity is considered at the boundaries.
27 For the deep water simulation, in the vertical direction, the computational nodes are
clustered near the upper boundary. The physical coordinate z is transformed by:
( =exp(Z-1-& ) where Z is a constant parameter which satisfies Z~L . The equations
are discretized in the transformed variable (using a staggered central-difference
scheme of second order to satisfy the momentum and density equations and to
calculate velocity and density perturbations.
For the shallow water simulation, in the vertical direction, the computational nodes
are uniformly clustered form the bottom to the top.
4.3 Domain of the simulations
In the deep water simulation, the domain was a 3D volume with the lengths being
respectively equal to l x l x 1.5 for X Y and Z. The objective was to simulate a flow in
shallow water so the length of the domain must be changed. Further, results would be
compared with data field so the shallow simulation must coordinate with these data.
The experiments data were taken in the Florida water where the depth was
approximately 20 meters. The mean strength of the wind blowing was 10.9 rn!s with a
mean direction of 170° from magnetic North. The wind sensor was located at 7
meters above the sea level, Large and Pond [15] method calculation gives a
corresponding wind blowing at a height of 10 meters having a speed of 11.25 m/s.
To determjne the domain of the simulation, we need to non-dimensionalize the data observed in experiments. The non-dimensional length scale is given by L = u. If
28 with u. being the friction velocity at the surface. And f is the Coriolis force equal to
f = 2QsinA . Because the Earth need 23 hours 56 minutes and 4 seconds to do a complete rotation on its axis (sidereal period), the Earth's rotation in rad I sec is
5 Q = 21l" 1(23 X 3600 +56 X 60 + 4) = 7.29 X 10" rad Is . At latitude 26°N, the Coriolis parameter is, hence, then f = 6.3933 ·10-5 rad Is .
The shear stress at the surface of the water is given by 7: = P water · u?. Because the shear stress across the free surface is continuous it can also be written as
r =Pair · C d · u ;ind . By substitution, the friction velocity at the surface is
u. = u wind (4-16)
3 3 with P water = 1024kg I m and P air = 1.3kg I m
To determine the drag coefficient factor C d at the surface, the Large and Pond[l5] method is used. For a wind speed at a height of lOrn blowing of 11 to 25 m/s, the following formula is given
(4-17)
1 With Uair =11.25m·s- we have C d =0.0012. Using Equation (4-16) we obtain a
1 friction velocity at the free surface equal to u. = 0.0139m · s - •
We are now able to determjne what should be the grid size of the simulated flow which should correspond to the experimental data. The depth of the flow observed in experiments was about 20 meters. To non-dimensionalize it, the depth is divided by the length scale giving
29 Z Depth = Depth· = 0.092 "" 0.1 length L L = u.
So for shallow water, the grid lengths are 1 for x, 1 for y and 0.1 for z. The number of points is 65 x 65 x 32 .
30 5 RESULTS
The code used for this research is the one that was developed by Zikanov et al.[l].
The code is written in Fortran and for this research it is compiled and run on a single
processor 600 MHz Compaq DEC XP-1000 professional workstation in a UNIX
environment. A first step of the research was to get familiar with this code, to be able
to run the simulation with the possibility of setting different parameters and compare
the results with corresponding ones obtained by Zikanov et al.. A sample computation
relevant to present problem, was carried out for the case with the parameters:
Latitude 26°N
Wind stress has an angle y = 45° with the South-North direction
The flow is assumed to be in deep Ocean. A free-slip boundary condition is
chosen at the bottom.
Data averaging interval has to be large enough to eliminate the possible impact of the slow fluctuations. Zikanov et al.(2002) showed that averaging over 5T1 (T1 being the non-dimensional inertial time period, T1 =2n) is accurate in presenting the first three statistical moments of the turbulent fluctuations. The following simulation data presented are time averaged from T=2T1 to T=7T1
31 1.5 ,---;:::----,.-----,.----==:c:====-.----~
' \
'I
N
0.5
~ ~ O L------~------~------~------L------~ -10 -5 0 5 10 15 horizontally averaged velocity
Figure 5-1 Horizontally and time-averaged profiles of the mean horizontal current u and v
Figure 5-l represents the vertical distribution of the components of the mean horizontal velocity of the flow. For each value of the depth Z, horizontal components of the velocity are horizontally and time averaged. Velocity profiles are similar to what Zikanov et al. found in their simulations.
In the figure 5-2, u and v are represented as a hodograph where the velocities at various depths are plotted. The line is showing the angle of the current with the wind direction. For this case, the simulation gives a current angle at the surface equals to
40.6° with the wind direction.
32 12 ~----~----.------,----~-----,------.-----.------.----,
10 I "'j \ I i \ I j 8 I j \ 1 wind direction 1 1 6
A ::>- v 4
2
0
2 0 -2 -4 -6 -8 -10 -12
Figure 5-2 Velocity hodograph
1 5 ~ ~--._ 1 5 '\ ) --- -...... ) 1'------} / ---- I ( \
N r-J
05 05
-- ·
Figure 5-3: Subgrid grade and Reynolds stresses
33 Figure 5-3 represents horizontally and time averaged profiles of vertical stresses.
subgrid-scale stresses. These profiles are similar to Zikanov et al.'s profiles, but not
exactly identical since this particular case was not run.
The effective viscosity is a function of depth and is evaluated using the computed
mean shear and the total stress profile as
(5-1)
Figure 5-4 represents the effective viscosity as a function of depth. Below the depth
of 0.5, it is not possible to evaluate the viscosity because of the small amplitude of
mean current and stress. As Zikanov et al. found, the viscosity grows rapidly with
depth and then decreases. Compared to their results, the effective viscosity doesn't
decrease monotonically and has more variations than what they found.
Results are generally quite similar to the Zikanov et al. results. Some differences
appear in some profiles. The flow having these particulars parameters have not been
simulated by Zikanov et al.. That could be the reason why small differences are
found.
34 1.5
1.4
1.3
1.2
1.1
N
0.9
0.8
0.7
0.6
0.5 1 2 3 4 5 6 7 8 9 10 Effective viscosity X 10-3
Figure S-4: effective viscosity coefficient
35 6 DISCUSSION
6. 1 Original simulated flow in shallow water compared with
deep water simulation
Shallow water simulations are simulated with latitude 26°N and a wind stress having an angle of 170° with the South-North direction. The flow is assumed to be in shallow water. Boundary stress at the bottom is applied following equations (3-16) and (3 -17). The following simulation data presented are time averaged over period
T=4T1 to T=12TI. In this part, shallow water simulation data are compared with a current flow having the same properties but simulated with characteristics corresponding to the deep water. For figures showing time-averaged data of deep
water simulation, the averaging range is 2T1 < t < 7T1 as in Zikanov et al.(2002).
Figure 6-1 shows the volume averaged kinetic energy of each velocity components as a function of time. At the beginning of the simulation, we can observe a sinusoidal oscillation. This oscillation is not really normal since the flow is turbulent. After time
5T1 , the kinetic energy of each component is not oscillating with sinusoidal profiles anymore.
36 Figure 6-2 shows the volume averaged kinetic energy of each velocity components as
a function of time of the deep water simulation. In this simulation, kinetic energy
profiles don't have any sinusoidal oscillation but only strong oscillations. Zikanov et
al.[l] supposed that theses strong oscillations could be of purely stochastic nature, or
can probably include a dominating inertial component correlating with the Earth
rotation.
80 I I 'I I I I 75 I ,, > I I 1\ I >v 70 I I I 65 I I '.t 2 4 6 8 10 12 14 16 18
> 1\ N ::J 5 v 0 2 4 6 8 10 12 14 16 18 t/T1 10 ~ 0 Figure 6-1: volume-averaged kinetic energy as a function of time for shallow water flow simulated at latitude 26 ·and wind direction being 110• from south-north direction 37 \./-\ > 8 \ II ~ \ "' ,..- ..... v 6 \ I \ I \ \ I \.. I ' I '--- ...... ,./ ' ,..-, __ ,/ ..,...,_~/' ·\ 0 I . ~ - . - . - ·- . - . - · - . - .--. - .--. - . --. - . --. -. -- . - . --. - . -- . - . - -.-. -- . - . 2 3 4 5 6 7 t/T1 4 ~ 3.5 0 ~ + "'2. 3 "0 0. *tJI 2.5 2 3 4 5 6 7 t/T1 Figure 6-2 volume-averaged kinetic energy as a function of time for deep water flow simulated at latitude 26 °and wind direction being 170° from south-north direction Kinetic energy values are more important in the shallow water case because data are volume averaged. The deep water simulation has a big volume where velocities are very low and from depth of about 0.5 to the bottom velocity components are equal to zero. The volume averaged result is thus decreased compared to the shallow water simulation. Figure 6-3 represents the horizontally ant time averaged vertical profiles of the flow velocities u and v of the shallow water simulation. The dotted lines represented in the same plot show the top part of the same velocity profiles but for the deep water 38 simulation. Figure 6-4 represents the whole vertical velocities profiles of the deep water simulation. At the surface, the u velocities of shallow and deep water simulations are similar while the v velocity of the shallow water is almost two times more important than in deep water. From the surface to the depth of the shallow simulation (0.1) the slope of each velocity are similar in the both simulation cases. 0 -0.01 -0.02 -0.03 -0.04 = .f -0.05 N -0.06 -0.07 -0.08 - shallow 1 -- Figure 6-3: Horizontally and time-averaged profiles of the components of the mean horizontal current u and v of shallow water at latitude 26 ·and wind direction being 170• from south-north direction 39 0 ' ' ' ' ' \ -0.5 a. =Gl "9 N -1 horizontally averaged velocity Figure 6-4 Horizontally and time-averaged profiles of the components of the mean horizontal current u and v of shallow water at latitude 26 oand wind direction being 170° from south-north direction Figure 6-5 shows the horizontally and time averaged speed of the current simulated in shallow water at latitude 26° and wind direction being 170° from South-North direction. The dotted line represented in the same plot shows the top part of the same speed profile but for the deep water simulation. Figure 6-6 represents the whole vertical speed profile of the deep water simulation. From the surface to the shallow depth, the current speed decreases faster for the deep water simulation than for the shallow simulation. The deep water flow speed becomes zero at a depth of about 1 from the surface. 40 0 ~ . 01 ~.02 ~.03 ~.04 :5 ,~ . 05 N ~ . 06 ~ . 07 ~ . 08 ~ . 09 -0.1 4 5 6 7 8 9 10 11 12 Speed Figure 6-5 Speed of current of the simulated flow for shallow water at latitude 26° and wind direction being 170° from south-north direction ~.5 -1 -1.5 L..____ _L______. ____ ..J...______l. ____ .L__ ___ I s _J I 0 2 4 6 8 10 12 Speed Figure 6-6 Speed of current of the simulated flow for deep water at latitude 26° and wind direction being 170° from south-north direction 41 The effective viscosity profiles of the two depth simulations are represented in the figure 6-7 and in the figure 6-8. In the shallow water, the viscosity increases from the surface to about the half of the depth. From the other half to the bottom depth, the viscosity decreases and become almost equal to zero at the bottom. The maximum is reached at z = Lz - 0.048 which is almost the middle depth of the flow. For the deep water simulation, the viscosity grows rapidly with depth until z = Lz - 0.3, then oscillate around its maximum and decreases from z = Lz - 0.5 to z = Lz -1. Below this depth, it is not possible to evaluate the viscosity because of the small amplitude of mean current and stress. 0 ,---,----,---,----,---,---,----,---.----.---, -0.01 -0.02 -0.03 -0.04 :5 ! -0.05 -0.06 -0.07 -0.08 -0.09 -0 . 1 ~~~--~--~----~--~--~----L---~---L~~ 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Effective viscosity Figure 6-7 Effective viscosity coefficient of the simulated flow for shallow water at latitude 26° and wind direction being 170° from south-north direction 42 -0.2 -0.3 -0.4 -0.6 -0.7 -0 .8 -0.9 - 1 L---~--~----~--_L ____L_ __ _L __ ~L---~--~----- o 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Effective viscosity Figure 6-8 Effective viscosity coefficient of the simulated flow for deep water at latitude 26° and wind direction being 170° from south-north direction Figure 6-9 and figure 6-10 show the vertical profiles of the root mean-mean-square velocity fluctuations for shallow water and deep water simulations_ In the shallow water case, the turbulent kinetic energy is higher near the surface and then decrease with depth. The u and v components of the kinetic energy re-increase from z = 0.015 to the bottom. The turbulent kinetic energy of the deep water simulation decreases from the surface to the bottom of the flow. 43 0.1 .... 0.09 / / / 0.08 I I I O.o7 I I 0.06 I N 0.05 I I I 0.04 I I 0.03 I I 0.02 - Figure 6-9 Root-mean-square velocity fluctuations of the simulated flow for shallow water at latitude 26° and wind direction being 170° from south-north direction / I I I / / / I / "/ j / N 0.5 1\ 0 / 0 0.5 1.5 2 2.5 3 rms velocity fluctuations Figure 6-10 Root-mean-square velocity fluctuations of the simulated flow for deep water at latitude 26° and wind direction being 170° from south-north direction 44 Figure 6-11 show the hodograph of the vertically averaged velocities of the shallow water simulation compared to the deep water simulation. In shallow water, obviously the flow is far from reaching the Ekman depth. The curve does not represent a spiral as it does for deep water. At the surface, the angle of direction of the current from the direction of the wind is equal to 45.8° for the shallow water model. For the deep water simulation, this angle was found to be equal to 33.5°. The angle found for the shallow water seems to be really too high compared to what we should obtain. 12 10 deep water 8 6 I I ~ II 4 :::) v I 2 I I I I / 0 "' '"' :"""-- -2 -4 5 0 -5 -10 -15 Figure 6-11 Velocity hodograph of the simulated flow for shallow water and deep water at latitude 26° and wind direction being 170° from south-north direction. 45 In the equation ofthe Drag coefficient Cd (Eq.3-15), some parameters are not exactly known. In his paper, Csanady was not sure about the Values of the constant B - Cr and the constant C; of equation (3-14). In order to see how it affects the flow, simulations with different values of the constant C; were done and compared with the previous shallow water simulation. Also, the depth was 0.1 in the shallow simulation. The both bottom and surface boundary conditions affects the flow with a lot of importance. In order to separate the two boundary actions in the flow, others simulations with larger depths were done and compared. 6. 2 Variation of drag coefficient simulations In this part, only the Drag coefficient parameter was changed to see and compare how it affects the flow. One current have been simulated with the coefficient C; multiplied by 2 from the original case, soC; = 21 .4, and one other current have been simulated with C; divided by 2, C; = 5.35. Figure 6-12 shows the variation of the drag coefficient with the time of the original shallow water simulation and of the two others simulations with the change of the drag coefficient value. Because the parameter C; is in the denominator of the drag coefficient equation, increasing it makes Cd smaller and decreasing C; makes the drag coefficient bigger. As we have seen it for the volume averaged kinetics energy and the current speed of the shallow water simulation (figure 6-1 ), there is a sinusoidal oscillation problem at 46 the beginning of the simulation. The current speed U near the surface is needed to calculate the drag coefficient (Eq.3-15) and this is why we must find the same kind of oscillations in both plots. X 10-3 f\ C.=5.35 / \ ...~~--- · ----...-·...... _,...,...._.....,.._ . / 3 ,/ I 2.5 - 2 1.5 . ···· ··· ...... ~ ...... 5 10 15 20 Figure 6-12 Variation of Drag Coefficients with the time for different values of the constant C; Figure 6-13 shows the volume averaged kinetic energy of each velocity components as a function of time of flows simulated with changes of drag coefficient. The flow having C; = 21.4 has a more important beginning part with a periodic oscillation than the one having C; = 5.35. The v component of the velocity is much more important 47 than the u component and is also much more concerned by the periodic oscillation phenomenon. C.=5.35 C.=21.4 I I 160 120 I 1-- ""l ""l 0 12 0 12 ~ ~ + 10 + 10 "'2- "'2- ~ 8 ~ 8 8. 8. Ul 6 Ul 6 2 4 6 8 3 5 7 9 11 13 t/T1 t/T1 Figure 6-13 volume-averaged kinetic energy as a function of time for shallow water flow simulated at latitude 26 o and wind direction being 170° from south-north direction with C,. = 5.35 (on the left) and C,. = 21.4(on the right). Figure 6-14 show the hodograph of the vertically averaged velocities of the original shallow water simulation compared with the flows simulated with changes of drag coefficient. We can clearly see that when the coefficient C,. is increased, and then the 48 drag coefficient Cd is decreased, the v component of the velocity becomes bigger and the angle of direction of the current from the direction of the wind becomes more important. When C; is decreased we have the contrary. The angle of direction of the current from the direction of the wind is equal to 43.9° at the surface when C; = 5.35 and is equal to 50.4° when C; = 21.4 . If th e drag coefficient needed to be changed, it seems that it would be more appropriate to increase it to obtain an angle of the current with the wind direction smaller from what we have. 10 Ci=10.7 8 / 6 I I 4 /\- ::::> v 1 2 Ci=5.35 Ci=21.4 I 0 -2 I I / / -4 I 5 0 -10 -15 Figure 6-14 Velocity hodograph of the simulated flow for shallow water with changes of the drag coefficient properties. 49 6.3 Variation of the depth Some simulations with different depths have been calculated and compared in this part. The first shallow water simulation was done with a depth equal to 0.1 which gives in a dimensional scale a depth of about 20 meters. A flow with the shallow water properties was simulated at a depth of 0.2 (about 40 meters) and another flow was simulated at a depth of 0.3 (about 60 meters). In the original shallow water simulation, the computational grid has 32 nodes in the vertical direction, but because the depth is increased for these simulations, the number of grid points was also increased to 64 instead of 32. This was done because with only 32 points in the vertical dimension, the simulation was not able to run properly and gives errors. Figure 6-15 represents the horizontally ant time averaged vertical profiles of the flow velocities u and v of the shallow water simulations having a depth of 0.1, 0.2 and 0.3 below the surface. Horizontal lines are drawn to represent the bottom depth of each simulation. The dotted line represents the top part of the velocities u and v of the deep water simulation. Values of u components of the three different depths are almost equals at the sea surface. They vary for each simulation in order to reach almost the same value again at their bottom. The u velocity is equal to zero at z = Lz - 0.057 for the flow having a total depth of 0.1, z = Lz - 0.085 for the flow having a depth of 0.2 and z = Lz - 0.119 for the flow having a depth of 0.3. Then the three shallow water flows have the velocity u being equal to a value around -5 at the bottom. In these simulations, a stress is applied at the bottom and then should be the reason of this increase of the u velocity. 50 Values of v velocities are similar for simulations with a depth of 0.2, 0.3 and the deep simulation. Value v is more important for the flow having a depth of 0.1 and slopes of decrease are similar for the 3 shallow water simulations and for the deep water simulation. 0 .I \ \ \ \ \ \ \ \ \ \ -0.05 \ \ \ \ \ \ \ \ \ -0.1 ' a~ -0.15 N -0.25 \ : \ \ -10 -5 0 5 10 15 horizontally averaged velocity Figure 6-15 Horizontally and time-averaged profiles of the components of the mean horizontal current u and v of shallow water simulation with a depth of 0.1, 0.2 and 0.3. Dotted line represent u and v of the deep water simulation. Figure 6-16 shows the horizontally and time averaged speed of the current simulated in shallow water simulations having a depth ofO.l, 0.2 and 0.3 below the surface. The 51 dotted line represents the top part of the current speed of the deep water simulation. Total speeds of every simulation except the one with a depth of 0.1 are nearly close at the sea surface while the current simulated with a depth of 0.1 is faster at the surface. Just below the surface, this current decreases slower than the three others that have more important depths. Simulations with depths 0.2 and 0.3 have a speed current which decrease until they reach a particular depth and then re-increase to the bottom. Figure 6-16 need to be observed with care, since the total current speed is the square root of the sum of the velocity components u and v, it can only be equals to a positive value and thus gives no idea of the direction of the current but only the amplitude of it. The minimum amplitude of the speed of the current simulated is almost at the bottom for the flow having a depth of 0.1 , at z = Lz - 0.135 for the flow having a depth of 0.2 and at z = Lz- 0.215 for the flow having a depth of 0.3. This speed acceleration close to the bottom could correspond to the boundary layer induced by the bottom stress. This phenomenon is not clearly visible in the simulation having a depth of 0.1 because the length between the surface and the bottom could be too close to each other. This could also be the reason why the current speed is a little bit faster at the surface than the two others simulations: the depth being very small, there is not as much water mass as the others simulations and thus the current is less slowed. The positive gradient of amplitude at the bottom of the flow simulated at a depth of 0.2 and 0.3 seems to come from the u velocity since this velocity re-increases near the bottom whereas v values of every simulation decrease from the top to the bottom. 52 -0.05 .c 15.. Q) ~ -0.15 -0.25 -O.J OL ___.l. 2 _____j 4-=::::::::=-6L____ _[ 8____ 1L 0 ____1J: 2=====...114 Speed Figure 6-16 Speed of current of the simulated flow for shallow water simulation with a depth of 0.1, 0.2 and 0.3. Dotted line represents the speed current of the deep water simulation. Figure 6-17 show the hodograph of the vertically averaged velocities of the original shallow water simulation with a depth of 0.1 compared with flows simulated with a depth of 0.2, 0.3 and the flow simulated in deep water. At the surface, the angle of direction of the current from the direction of the wind is equal to 45.8° for the simulation having a depth of 0.1, 30.7° for the simulation having a depth of 0.2 and 28.9° for the simulation having a depth of 0.3. For the deep water simulation, this angle was found to be 33.5°. 53 As it was already said, the angle at the surface of the flow having a depth of 0.1 seems to be too high. The angle of the flow having a depth of 0.2 is still larger than the one of the deep water simulation. Only the current simulated with a depth of 0.3 has a surface angle smaller than the deep water simulation with a similar speed. We can clearly see that that for flow simulated with depths of 0.2 and 0.3, speeds re- increase at the bottom of curves which correspond to the bottom of the simulated flows. Near the surface, deeper simulated flows are less sensitive to the bottom effects and thus have a curve corresponding more to the one simulated in deep water. 12 depth 0.3 depth 0.2 10 r: I j: I depth 0.1 j I 8 .: I I I !: I r: l 6 j .· I . · I I · . I 4 _I I II- :::1 .r I v l I 2 / I I I .i deep 0 ...... ·- .... ,,_-;...... / - j .' - . . - .. .-. / I I / I -2 / I / I I I -4 I I I I I I 5 0 -10 -15 Figure 6-17 Velocity hodograph of the simulated flows for shallow water with changes of the depth and simulated flow in deep water. 54 6.4 Ekman transport 6.4.1 Mass transport in deep water The total mass transport of the simulated flow is given by (see Background) 0 M x = fpudz - Z 0 M Y= fpvdz - Z In the simulated case, the data are non-dimensionalized. The typical velocity scale is the surface friction velocity u • = ~ so we have u = u'u • and v = v'u • with u' and ~Po v' being the non-dimensionalized components of the velocity. The length scale is the turbulent length L = u • I f so we have z = z' · u • I f . With the stress tensor being *2 equal to z-0 = pu the mass transport components become 0 M = 'fo Ju'dz' x f - Z' 0 M =~ Jv'dz' Y f -z· Non-dimensional mass transport is given by M' = fMx = ofu'dz' X 1" 0 - Z' M ~ = JMY = Jv'dz' -r o -z· 55 Ekman assumed the Total volume transport to be equal to so r -r M =--" M' = ---' = cos y y Y r f 0 In the simulated case, the wind is blowing along the x-axis. The non dimensional mass transport components should be equal to M'X =0 M'y = -1 To compare these results with the simulation, velocities were averaged with respect to the horizontal coordinates and time with time-averaging being over period 5T,. Figure 6-18 represents the profiles of the mean horizontal x and y components of velocity. We can clearly observe the rotation and the reduction of the velocity with the increase of the depth. The simulation was computed in deep water, at latitude 26° North with a wind stress having an angle of 170° with the South-North direction. The computed non-dimensional mass transport found were M: = 0.005 and M ~ = - 1.1608 . Compared to the theory that predicted M; = 0 and M ~ = -1 , the simulation results are really similar. Both show that the net mass transport is to the right of the applied surface stress. But the result does not depend on the eddy viscosity assumption because the equation Jvdz = -r I fp follows directly from a vertical integration of the equation of motion in the form - pfv = dr I dz. 56 wind direction 1.5 0.5 0 10 14 0 -5 4 2 -10 -2 X y Figure 6-18 Perspective view showing velocity decreasing and rotating with increase of depth of deep water flow simulated at latitude 26° with a wind stress having an angle of 170° with the South-North direction 6.4.2 Mass transport in shallow water The mass transport was also computed from the shallow water simulation. Velocities were averaged with respect to horizontal and time, with time-averaging over period • 4T1 to l2T1 Figure 6-19 represents the coordinate profiles of the mean x and y components of velocity. In this case, the depth is not big enough to reach the Ekman depth where the current is opposite to the surface current direction. The computed non-dimensional mass transport found were M x = 0.0958 and M Y = -0.8265. 57 wind direction 0.1 0.08 0.06 ~.04 0.02 0 10 5 15 10 0 -5 y -10 -5 X Figure 6-19 Perspective view showing velocity decreasing and rotating with increase of depth of shallow water flow simulated at latitude 26° with a wind stress having an angle of 170° with the South-North direction 6.5 Comparison with the field observations The experiments data come from a current observed in the South Florida Ocean by an ADCP mounted on the sea floor at a depth of about 17 meters from the surface. Experiments were made in April 200 by a 150 kHz bottom-mounted Harbor Surveillance Current profiler, ADCP located at 26°5.7' N and 80°5.2' W. A low- pressure atmospheric front was observed as it passed through the Dania Beach site on April 9, 2000. Figure 6-20 shows the wind vectors during the front passage. 58 Horizontal velocities of the current were observed during a period of almost 6 hours where the wind current was observed to be constant in direction and magnitude. The measurement of the wind properties was done by an Air Sea Interaction Spar (ASIS) Buoy. The ASIS is a multi-sensor platform used to measure directional wave-spectra, air-sea surface fluxes and radiation. During the experiments of April 2000, the ASIS buoy was moored at location 26°2.4'N and 80°5.43'W. The ASIS wind sensor was located at 7 meters above the mean sea level. During the front passage, the mean strength of the wind blowing was 10.9 m/s with a mean direction of 170° from magnetic North. 26 Averaging Period (if ' ..... ~ ..._,, , ,. 0 1 ]_ ;ASJS8uoy . /_.,.., .'l· - }_· ·",1 -26 1 8 8 .5 9 9.5 10 10.5 11 11.5 12 Figure 6-20 Wind vectors from meteorological station displaying change in wind strength and direction as low-pressure cold front passes through region (M. Chernys Thesis [171). Figure 6-21 shows the velocity contours in the local water column during the experiment period measured in meter/second, and the depth averaged North and East components of current during the days before and after the passage of the front. It is visible that before April 9, the north component of velocity was mostly positive throughout the column. In April 9, when the wind shifted direction from southwest to 59 northwest, the current direction shifted with the north component and then became negative. 15 E ~10 ii Ill 2 0 5 4 75 8 8.5 9 9.5 15 ~ m/s ;- 10 Q. 8 5 2 9 04 7j)e 7.5 8 8.5 9 95 10 10 5 11 Day {April 2000) Figure 6-21 Ocean current contours as recorded by the bottom-mounted ADCP measured in meter/seconde. Top panel represents North component and middle panel represents East component of velocity. Bottom panel shows the depth-averaged North (solid line) and East (dashed line) components over the experiment period (M. Chernys Thesis [171). But the shift of direction of current was visible about 2 hours before the shift in wind direction. The reason of this could be because current was generated at northern latitude where the shift direction of wind was visible earlier [ 17]. During the shift in current velocity in the water column observed in April 9, the winds and ocean currents were averaged over a period of 5.76 hours that signified a period 60 nearly constant wind strength and direction. Figure 6-22 shows a perspective view of the mean horizontal velocity in x and y directions for each depth measured by the bottom-mounted Harbor Surveillance Current profiler. -. · - · - -..wind direction . ·- ·- .- .- .- -- ~ -- == ;-;;~, N 4 2 0.2 -0.1 y X Figure 6-22 Perspective view showing velocity decreasing and rotating with increase of depth of the observed flow The angle of direction of the current from the direction of the wind is equal to 20.1 o at 1 meter depth and 25.2° at 16 meter depth. Because of the reflection effects of the surface, values measured by the ADCP near the water surface are not usable. 61 Figure 6-23 represents the hodograph of the vertically averaged velocities of the original shallow water simulation with a depth ofO.l compared with flow observed in experiments. In order to compare the both flows, the observed values are non- dimensionalized. The velocity components are divided by the typical velocity scale which is the surface friction velocity that was found to be u. = 0.0139m · s -1 for a wind blowing at 10.9 m/s. 16 field observations 14 12 A I I \ I I \ 10 I \ I \ 8 1\ ::::>- v 6 4 2 0 wind direction -2 -4 5 0 -5 -10 -15 Figure 6-23 Velocity hodograph of the simulated flow for shallow water and of the observed flow 62 Profiles obtain from experiments and from simulation show that there are similarities in variations of the curves. But angles and amplitudes of the two flows at every depth are completely different. The shift in the direction of current began about 2 hours before the observed shift in wind direction. Chemys [ 17] suggests that this is due to the generation of the current by the wind at locations north of the observed flow first. Also, others influences in the region include the Florida Current. But during the passage of the observed front, Chemys found with the help of Ocean Surface Current Radar (OSCR) images that the influence of the Florida Current in the littoral zone was negligible. One other problem with these observed data is the period of the wind observation. Only a period of about 6 hours with a constant wind direction and strength was observed. Even if this flow was generated in shallow water and then should need less time to become steady than a wind driven current in deep water, the total observation period seems to be too short to conclude the flow to be an Ekman current. 63 7 CONCLUSIONS AND FUTURE WORK Ekman [2] developed a mathematical theory of a current induced in deep water by wind through a friction force at sea surface and subjected to the Coriolis force. Assumptions made in this model were later contested. To investigate the Ekman flow in detail, Zikanov et al. [1] developed a numerical model using a Large Eddy Simulation method to simulate the wind driven flow. As an extension to Zikanov et al., this work considers an Ekman wind driven flow generated in shallow water instead of deep water. To compare with field observations obtained from an experiment, the simulated flow was also adapted to the physical parameters corresponding to those of the experiment. For deep water simulation, a free slip boundary condition is applied at the bottom of the computational domain. For shallow water, a bottom stress is defined using a drag coefficient parameter and is specified as a boundary condition at the bottom of the computational domain. Comparisons are made with the deep-water simulation. The bottom boundary condition is not known accurately and, in order to determine a correct model for the shallow water simulation, the effect of the variation of the parameters in the bottom boundary condition is investigated. Essentially, the choice of the bottom boundary condition leads to an adverse flow close to the bottom, whose magnitude depends on the choice of the parameters in the bottom boundary conditions. In all of the simulations 64 attempted, some adverse flow is present. It is believed that this can be optimized, and through comparison with observation, used to determine an appropriate bottom boundary condition. Each simulation run, however, takes a number of days to run and requires period beyond what is available for this thesis work to complete. This is therefore left as part of future work. The results obtained, however, are useful in making comparison between different depths and in comparing with the deep-water case. At the surface, the angle of the current with the wind direction was found to be equal to 45.8° for the simulation having a depth of 0.1, 30.7° for the simulation having a depth of 0.2 and 28.9° for the simulation having a depth of 0.3 corresponding to a dimensional depth of 20, 40 and 60 meters. For the deep water simulation, this angle was found to be33.5° at the surface. When the drag coefficient parameter is changed, the simulated flow has different properties. In shallow water simulation, the viscosity was found to increase from the surface to about the half of the depth and then re-decrease from the other half to the bottom depth and become almost equal to zero at the bottom. A computed non-dimensional mass transport was found to be equal to M_: = 0.0958 and M;, = -0.8265 for the shallow water simulation and M: = 0.005 and M ~ = -1.1608 for the deep water simulation. The simulated flow is compared with field observations. It was found that the current profiles are quite similar but the magnitude and direction of current are different. The differences may partly be due to the choice of the bottom boundary conditions. It may also be due to the fact that in the observations, the wind was blowing with a constant direction and amplitude only for a period of few hours. 65 Future work would be to identify what produced the sinusoidal oscillations of the velocity profiles as functions of time in the shallow water cases, determine why the angle of the simulated flow is so high when the depth is relatively small. To do that, variations of the drag coefficient or of the bottom stress conditions can be investigated and simulated. Also a comparison with a wind driven flow observed for a longer time in shallow water could be investigated. 66 APPENDIX Mathematical Solution ofthe Ekman equations: The simplified equations of motions are: (1-1) (1-2) with the following boundary conditions: au T x =ptA - Z = 0 (1 -3) z oz ' 8voz = 0 ' z= O (1-4) u = v = 0 , at z = - oo (1-5) To solve theses equations, a new variable known as the complex velocity, u + iv is considered. A single differential equation in the complex velocity can be formed by adding (1-1) to i x (1-2): o2 (u+iv) f(u+iv)+iAz = 0 (A-1) oz 2 in terms of complex velocity, the boundary conditions become r x= pA z~oz (u+iv), z =O (A-2) u + iv = 0 , z = -oo (A-3) 67 Solution in term of complex variable: If we let u + iv = cerz in equation (A-1 ), we have r = ±~if I Az then the general solution to (A-1 )is (A-4) Now if we apply the boundary conditions at z = -oo, Equation (A-3), yields c 2 = 0 for f > 0. The boundary condition at z = 0, Equation (A-2) then yields c1 = (rx I pBJ~B z I if. The solution in terms of the complex velocity is then . rx ifi / 8 . u + tv= --pj--ze z - (A-5) pBz if Transformation back to real variables: To get a solution for u and v separately requires use of the density e;e = cosB + isin B . (A-6) This mean we need to get i out of the square root sign in equation (A-5). We can use the density with e = TC I 4 to show that Ji = (1 + i) I J2. 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