A non-hydrostatic numerical model for calculating free-surface ... - ocean

Ocean Dynamics (2003) 53: 176–185 DOI 10.1007/s10236-003-0039-6

Yuliya Kanarska
Vladimir Maderich

A non-hydrostatic numerical model for calculating free-surface stratified flows

Received: 31 October 2002 / Accepted: 3 June 2003
Springer-Verlag 2003

Abstract A three-dimensional non-hydrostatic numerical model for simulation of the free-surface stratified flows is presented. The model is a non-hydrostatic extension of free-surface primitive equation model with a general vertical coordinate and horizontal orthogonal curvilinear coordinates. The model equations are integrated with mode-splitting technique and decomposition of pressure and velocity fields on hydrostatic and nonhydrostatic components. The model was tested against laboratory experiments on the steep wave transformation over the longshore bar, solitary wave impact on the vertical wall, the collapse of the mixed region in the thin pycnocline, mixing in the lock-exchange flows and water exchange through the sea strait. The agreement is generally fair. Keywords Numerical modelling
Non-hydrostatic pressure
Free surface flows

1 Introduction The 3-D hydrostatic free-surface primitive equations are used in most numerical models of estuaries, coastal seas and ocean circulation. In the case of constant density an equivalent set of equations is called shallow water approximation. However, non-hydrostatic effects remain important in a wide spectrum of stratified flows even when the horizontal scale of the process is much more than vertical scale. The examples are steep waves on uneven bottom in the coastal areas (Beji and Battjes 1994), the buoyant plumes from submerged outfalls, breaking of internal waves of large amplitude generated Responsible Editor: Hans Burchard Y. Kanarska (&)
V. Maderich Institute of Mathematical Machine and System Problems, Glushkov av. 42, Kiev, 03187, Ukraine e-mail: [email protected]

by the tidally driven flows over a steep topography (Farmer and Armi 1999) and exchange flows over the sills in sea straits (Zhu and Lawrence 1999). The nonhydrostatic formulation of the problem is necessary for the simulation of deep convection events in the open ocean (Marshall and Schott 1999). Unlike the hydrostatic approximation, the non-hydrostatic formulation of the problem is well posed in a domain with open boundaries (Mahadevan et al. 1996a). In recent years, several numerical models have been developed to simulate the above-listed non-hydrostatic geophysical flows. In Table 1 the major characteristics of a number of models are given. A straightforward numerical solution of full Reynolds-averaged Navier–Stokes (RANS) equations to determine unsteady three-dimensional fields of velocity, pressure and scalars and a twodimensional field of free surface elevation is much too computationally expensive for most of these flows. To improve the efficiency of the method in several models, the pressure and velocity fields are decomposed into hydrostatic and non-hydrostatic counterparts and components are found sequentially. In the first step, Mahadevan et al. (1996b), Casulli and Stelling (1998) and Casulli (1999), for example, neglect the contribution of the non-hydrostatic pressure and velocity. The surface elevation and the provisional velocity are obtained implicitly. At the second step, the provisional velocity is corrected by including the non-hydrostatic pressure terms in such a fashion that the resulting velocity field was non-divergent (Harlow and Welch 1965). Casulli (1999) included also a non-hydrostatic correction for the free surface. However, the developed approaches are not compatible with the class of hydrostatic ocean model that uses mode-splitting and terrain-following vertical coordinates (POM: Blumberg and Mellor 1987; Ezer et al. 2002; GHER 3D: Beckers 1991; SCRUM: Song and Haidvogel 1994; ROMS: Haidvogel et al. 2000; Ezer et al. 2002; POL3DB: Holt and James 2001; GETM: Burchard and Bolding 2002). In this paper a three-dimensional numerical model for simulation of the unsteady free-surface density stratified flows is presented. Unlike other non-hydrostatic models,

177 Table 1 Non-hydrostatic models Reference

Upper boundary

Baroclinicity

Vertical coordinate

Horizontal coordinate

Mode splitting

Pressure decomposition

Mahadevan et al. (1996) Marshall et al. (1997) Casulli and Stelling (1998) Kampf and Backhaus (1998) Stansby and Zhou (1998) Casulli (1999) Hodges and Street (1999) Rasmussen et al. (1999) Winters et al. (2000) Stelling and Busnelli (2001) Lin and Li (2002) This model

Free surface Free surface Free surface Rigid lid Free surface Free surface Free surface Free surface Rigid lid Free surface Free surface Free surface

Yes Yes Yes Yes No No No Yes Yes No No Yes

z and r z z z r z Curvilinear z Curvilinear z r s

Curvilinear Spherical Cartesian Cartesian Cartesian Cartesian Curvilinear Cartesian Curvilinear Cartesian Cartesian Curvilinear

No No No No No No No No No No No Yes

Yes Yes Yes No Yes Yes No No No Yes No Yes

2-D depth-integrated momentum and continuity equations (external mode) are integrated explicitly, whereas 3-D equations (internal mode) are solved implicitly with time-splitting technique for internal and external modes (Blumberg and Mellor 1987). The general vertical coordinate and orthogonal curvilinear horizontal coordinates are used in this model while the r-coordinate and Cartesian coordinates are used in a preliminary version of it (Kanarska and Maderich 2002). The model is based on the POM code, but it can be considered as non-hydrostatic extension of most ocean free-surface models with mode splitting. The paper is set as follows. In Section 2 the equations of the model, along with transformation of the equations to the general vertical coordinate, are presented. The numerical implementation and step-by-step algorithm are described in Section 3. Geophysically motivated applications and a comparison with laboratory experiments are given in Section 4.

2 Model The model is based on the 3-D RANS equations and the Boussinesq approximation. The equations in a Cartesian coordinate system ðx; y; z; tÞ are: oU oU 2 oUV oUW 1 oP þ þ ¼ þ ot oy oz q0 ox ox
2  o U o2 U o oU Þ þ fV ; þ 2 þ ðKM þ AM 2 ox oy oz oz oV oUV oV 2 oVW 1 oP þ þ ¼ þ ot ox oz q0 oy oy
2 2  o V o V o oV Þ  fU ; þ AM þ 2 þ ðKM ox2 oy oz oz

ð1Þ

ð4Þ

where U ðx; y; z; tÞ, V ðx; y; z; tÞ, W ðx; y; z; tÞ are the velocity components in the horizontal x, y and vertical z directions, respectively, P ðx; y; z; tÞ is the pressure, q is the density, q0 is the reference density, g is the gravitational acceleration, f is the Coriolis parameter, AM and KM are the horizontal and vertical turbulent eddy viscosity, respectively. The density q is determined from the seawater state equation q ¼ q ðT ; S; P Þ (Mellor 1991). Here T is temperature and S is salinity. The transport equation for scalar /i (e.g. /1 ¼ T ; /2 ¼ S) is: o/i oU /i oV /i oW /i þ þ þ ot ox oy oz
2 2  o /i o /i o o/ þ 2 þ ðKH i Þ þ SSi ; ¼ AH oz ox2 oy oz

ð5Þ

where AH and KH are the horizontal and vertical turbulent diffusivity. SSi denotes internal sources and sinks of tracer /i . We use two-equation k  l turbulence model (Mellor and Yamada 1982) to calculate the eddy viscosity and diffusivity. Here, for simplicity, the horizontal eddy diffusivity coefficients AH and AM are assumed to be constant; but the numerical algorithm of the present model is general and we consider the choice of turbulent model as independent option. The pressure P is decomposed into the sum of hydrostatic Ph and non-hydrostatic Q components as P ¼ Ph þ Q. The hydrostatic pressure component is determined from the vertical momentum Eq. (3) by neglecting the convective and the viscosity terms as:

ð2Þ Ph ðx;y;z; tÞ ¼ Pa þ gq0 gðx;y;tÞ þ g

2

oW oUW oVW oW þ þ þ ot ox oy oz 1 oP g  ðq  q0 Þ ¼ q0 oz q0
2  o W o2 W o oW Þ ; þ 2 þ ðKM þ AM 2 ox oy oz oz

oU oV oW þ þ ¼0 ; ox oy oz

Z0

q0 ðx;y;n;tÞdn;

ð6Þ

z

ð3Þ

where q0 ¼ q  q0 . For simplicity, only: the Cartesian horizontal coordinates are considered while the numerical algorithm was implemented in the horizontal curvilinear coordinate system. The general vertical coordinate (Burchard and Petersen 1997; Pietrzak et al. 2002) allows the bottom

178

relief to be represented smoothly and effectively resolves the bottom and the surface turbulent layers. The transformation of Eq. (1)–(4) written in the Cartesian coordinates (x ; y  ; z; t ) into equations in the new general vertical coordinates (x; y; s; t) results in the governing equations in the s-coordinate system as follows:
 oUJ og oQJ oQA1 þ AdvðUJ Þ ¼ gJ  BðxÞ ðq0 ; J Þ  þ ot ox ox os
 o KM oU þ fVJ ; ð7Þ þ þ AM Dif ðU Þ þ os J os
 oVJ og oQJ oQA2 ðyÞ 0 þ AdvðVJ Þ ¼ gJ  B ðq ; J Þ  þ ot oy oy os
 o KM oV  fUJ ; ð8Þ þ AM Dif ðV Þ þ os J os
 oWJ oQ o KM oW þ AdvðWJ Þ ¼  þ AM Dif ðW Þ þ ; ot os os J os ð9Þ where s ¼ sðx; y; z; tÞ is the generalized vertical coordinates with sðgÞ ¼ 0 and sðH Þ ¼ 1; J ¼ oz=os; A1 ¼ ðoz=oxÞs ; A2 ¼ ð oz=oy Þs ; A3 ¼ ðoz=otÞs ;

Here x1 ¼ x; x2 ¼ y. Dif is the horizontal diffusion operator, which transformed in new coordinates as: o oF o oF o oF o A2i oF þ Ai ; J þ Ai þ oxi oxi oxi os os oxi os J os i ¼ 1; 2 : ð10Þ

The transformed vertical velocity x is: x ¼ W þ UA1 þ VA2 þ A3 :

ð11Þ

The continuity equation in the new coordinate system is oUJ oVJ oUA1 oVA2 oW þ þ ¼0 : þ þ ox oy os os os

og og og þV þ ; ox oy ot

W ¼ U

oH oH V ; ox oy

s¼0

ð14Þ

s ¼ 1 :

ð15Þ

The boundary conditions for Eqs. (7)–(9) and Eq. (13) at s ! 0 are:
 KM oU oV ; ¼ ðs0x ; s0y Þ ; ð16Þ os os J
 KH o/i ð17Þ ¼ Hi ; J os where ðs0x ; s0y Þ are the surface wind stress components, Hi is the scalar flux rate. The corresponding boundary conditions at the bottom s ! 1 are
 KM oU oV ; ¼ ðsbx ; sby Þ : ð18Þ os os J The bottom stress is specified as sby ¼ CD jVb jVb ; 
 ! 1 zb þ z0 2 : CD ¼ max 0:0025; ln j z0

s

Dif ðF Þ ¼

W ¼U

sbx ¼ CD jUb jUb ;

oFUJ oFVJ oF x þ þ ; ox oy os  Z0  0 gJ oq os oq0 ¼ J þ ds0 : qo oxi oxi os0

AdvðFJ Þ ¼ Bðxi Þ

The kinematic boundary conditions at the free surface (s ¼ 0) and at the bottom (s ¼ 1) are:

ð12Þ

The equation for scalar /i ðx; y; s; t) is written as:
 o/i J o KH o/i þ Advð/i J Þ ¼ AH Dif ð/Þ þ þ JSSi : ot os J os ð13Þ The coefficients of transformation in the r-coordinate are
 oD og þ ; J ¼ D; D ¼ H þ g; A1 ¼  r ox ox

 oD og oD og þ ; A3 ¼  r þ : A2 ¼  r oy oy ot ot

Here Ub ; Vb are the corresponding velocities at distance from bottom zb in the layer of constant stress, z0 is the roughness length, j is the von Karman constant. A zero flux condition for temperature and salinity is imposed at the bottom. The normal and tangential velocities and scalar fluxes are set to zero at the solid boundaries. At the open boundaries the radiation conditions are used (Orlanski 1976; Palma and Matano 1996). The equations describing the dynamics of sea circulation contain fast external gravity waves and slow internal gravity waves. It is desirable to separate the vertically integrated equations (external mode) from the equations for vertical structure (internal mode). By applying kinematic condition at the free surface Eq. (14) and bottom Eq. (15), the depth-integrated continuity equation gives a free surface equation as: oU
J oV
J og þ þ ¼0 : ox oy ot

ð19Þ

Integrating the hydrostatic momentum equations (Q ¼ 0) over the depth from s ¼ 0 to s ¼ 1 with the boundary conditions Eqs. (16), (18), yields oU
J og þ AdvðU
J Þ ¼ gJ  BðxÞ ðq0 ; J Þ ot ox þ AM Dif ðU
Þ þ sbx  s0x þ f V
J þ Gx ;

ð20Þ

179

oV
J og þ AdvðV
J Þ ¼ gJ  BðyÞ ðq0 ; J Þ ot oy þ AM Dif ðV
Þ þ sby  s0y  f U
J þ Gy ;

ð21Þ

where AdvðFJ Þ ¼ Bðxi Þ ¼

Z0

oF U
J oF V
J þ ; ox oy

ð22Þ

The density qnþ1 is determined from the state equation using values of temperature T nþ1 and salinity S nþ1 :

Bðxi Þ ds; x1 ¼ x; x2 ¼ y ; Stage 2: free surface elevation

1

ðU
; V
Þ ¼

ð/i J Þnþ1  ð/i J Þn1 þ Advð/i J Þn 2DtI
 o o/nþ1 n1 i
KH þ JSSin : ¼ AH Dif ð/i Þ þ os os

Z0

ðU ; V Þds :

1

Gx and Gy are the so-called dispersion terms, which are the result of integrating over the depth the advective and diffusive terms in Eqs. (7) and (8) (see Blumberg and Mellor 1987). The 2-D system of Eqs. (19)–(21) describes external mode dynamics in the ‘‘shallow water’’ approximation.

3 Numerical algorithm In this section, the numerical algorithm of solving the governing equations with the boundary conditions is presented. The discretization by the finite differences method on the staggered Arakawa-C grid for spatial coordinates is used. The basis of numerical implementation of the present model is the decomposition of the velocity and pressure into hydrostatic and non-hydrostatic components and effective sequential calculation of these components. The numerical solution of governing equations is derived at four-stage procedure: (1) scalar fields calculations; (2) calculations of the free surface level and the depth-integrated velocity field; (3) calculations of the provisional hydrostatic components of velocity and pressure fields; (4) calculations of non-hydrostatic components of the velocity and pressure fields. The solution procedure of model equations on time interval [n  1; n þ 1] with external DtE and internal time step DtI are summarized as follows

At the second stage, the external mode Eqs. (19)–(21) is calculated on interval [n; n þ 1] with time-step DtE . In semidiscrete form they are: gmþ1  gm1 oðU
J Þm oðV
J Þm þ ¼0 ; þ 2DtE ox oy

ð23Þ

ðU
J Þmþ1  ðU
J Þm1 þ AdvðU
J Þm 2DtE ogm þ f ðV
J Þm þ AM Dif ðU
m1 Þ ¼ gJ m ox  BðxÞ ðq0 nþ1 ; J n Þ  sn0x þ snbx þ Gnx ;

ð24Þ

ðV
J Þmþ1  ðV
J Þm1 þ AdvðV
J Þm 2DtE ogm  f ðU
J Þm þ AM Dif ðV
m1 Þ ¼ gJ m oy  BðyÞ ðq0 nþ1 ; J n Þ  sn0y þ snby þ Gny ;

ð25Þ

where index m is corresponded to the external time step. The initial 2D velocity field at n-step is determined by the direct integrating of the general non-hydrostatic 3-D velocity fields from the n-internal time-step as R0 ðU
n ; V
n Þ ¼ 1 ðU n ; V n Þds. Then the depth integrated Eqs. (19)–(21) are solved with a short time step DtE . The obtained DtI =DtE fields of the free surface elevation are averaged at interval ðn; n þ 1Þ. The averaged value ~g is used in internal mode equations at the following stage. Thus two modes are directly coupled at each internal step.

Stage 1: scalar fields

Stage 3: hydrostatic components of velocity and pressure

The scalar fields (temperature, salinity, turbulent quantities) are computed semi-implicitly. The advective terms are determined using non-hydrostatic velocity fileld at the previous internal step n. The implicit treatment of the vertical diffusion terms is used. The obtained threediagonal system is solved by a direct method. The advective terms in equations for scalar function is approximated by the high order scheme (Smolarkewicz et al. 1998).

Following Casulli and Stelling (1998) the 3-D hydrodynamic equations without non-hydrostatic pressure Q ¼ 0 are solved semi-implicitly to determine the provisional values of velocity U~ ; V~ ; W~ . The advective terms and Coriolis acceleration are determined from the nonhydrostatic solution at the previous internal step n. The implicit treatment of the vertical diffusion terms is used. The momentum equations in the semi-discrete form with internal time step DtI are:

180

ðU~ J Þnþ1  ðUJ Þn1 þ AdvðUJ Þn 2DtI o~g ¼ gJ  BðxÞ ðq0nþ1 ; J n Þ þ f ðVJ Þn ox
 o KM oU~ nþ1 n1 ~ ; þ AM Dif ðU Þ þ os J os


2DtI

þ ð26Þ

ðV~ J Þnþ1  ðVJ Þn1 þ AdvðUJ Þn 2DtI o~ g  BðyÞ ðq0nþ1 ; J n Þ  f ðUJ Þn oy
 o KM oV~ nþ1 n1 ~ ; þ AM Dif ðV Þþ os J os



ð27Þ

ð28Þ

Stage 4: non-hydrostatic components of velocity and pressure The 3-D non-hydrostatic components of velocity are computed by correcting the provisional velocity field with the gradient of non-hydrostatic pressure Q. In the semi-discrete form they are " # ðUJ Þnþ1  ðU~ J Þnþ1 oðQJ Þnþ1 oðQA1 Þnþ1 þ ¼ ; ð29Þ 2DtI ox os " # ðVJ Þnþ1  ðV~ J Þnþ1 oðQJ Þnþ1 oðQA2 Þnþ1 þ ; ð30Þ ¼ 2DtI oy os ð31Þ

The continuity equation is satisfied by the final 3-D non-hydrostatic velocity field as oðUJ Þnþ1 oðVJ Þnþ1 oðUA1 Þnþ1 þ þ ox oy os þ

oðVA2 Þnþ1 oW nþ1 þ ¼0 : os os

nþ1

oðU~ J Þ oðV~ J Þ oðU~ A1 Þ oðV~ A2 Þ oW~ þ þ þ þ ox oy os os os

nþ1

ð33Þ

At this stage, all boundary conditions for the velocity field are satisfied. The obtained three-diagonal system is solved by the direct method.

ðWJ Þnþ1  ðW~ J Þnþ1 oQnþ1 : ¼ 2DtI os

o2 QJ o2 QA2 o A2 oQJ þ þ oy 2 os J oy oyos

o 1 oQ o A1 oQA1 o A2 oQA2 þ þ þ os J os os J os os J os ¼

¼ gJ

ðW~ J Þnþ1  ðWJ Þn1 þ AdvðWJ Þn 2DtI
 o KM oW~ nþ1 n1 ~ : ¼ AM Dif ðW Þþ os J os

o2 QJ o2 QA1 o A1 oQJ þ þ ox2 os J ox oxos

Lin and Li (2002) showed that the matrix of discretized Poisson equation in the r-coordinate and for regular rectangular discretization of the domain is reduced to a symmetric positive definite 19-diagonal matrix with 12 arisen from cross-differentiations due to the r-coordinate transformation. In the general case of the curvilinear orthogonal horizontal coordinates and irregular domain discretization we obtain a non-symmetric 15-diagonal linear system. Eight diagonals arise from the s-transformation components. The linear system was efficiently solved by the biconjugate gradient method with incomplete LU decomposition preconditioning (Seager 1988). The boundary conditions are satisfied at the second stage. Therefore from Eqs. (29)–(31), the condition of zero normal flow is imposed at the solid boundaries, which translates to the Neumann type of the boundary condition for non-hydrostatic pressure component Q. At the free surface and at the open boundaries Q ¼ 0. Once the hydrodynamic pressure is computed, the corresponding velocity field U nþ1 ,V nþ1 , W nþ1 is determined from Eqs. (29)–(31).

4 Applications In this section the geophysically motivated applications are presented and the comparison with laboratory experiments is given. The calculations were carried out with molecular viscosity and diffusivity. The r-coordinate was used as particular case of s-coordinates. The first three examples deal with the nonbreaking surface waves in coastal area, whereas the last three examples belong to the gravity currents class. 4.1 Short waves in deep rectangular basin

ð32Þ

By substituting the expressions for final velocities (29)–(31) into continuity Eq. (32), the Poisson equation for the non-hydrostatic pressure component Q is derived as

In this example the model is applied to the 3-D problem of short wave oscillations of small amplitude in a rectangular basin of length Lx ¼ 10 m, width Ly ¼ 10 m and depth H ¼ 10 m. Since the hydrostatic approximation describes long non-dispersive waves, it is not applicable for aspect ratio H=Lx ¼ 1. The analytical solution of the linearized 3-D problem (Kochin et al. 1964) is:

181

gðx; y; tÞ ¼

1 X 1 X

bnm cos rnm t cos

n¼0 m¼0



 px py n cos m ; Lx Ly ð34Þ

where bnm are coefficients of the decomposition of initial distribution of elevation, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rnm ¼ gsnm tanhðHsnm Þ;
2
2 pn pm s2nm ¼ þ : ð35Þ Lx Ly The computations were carried out on a grid with resolution 20 · 20 · 20 and with equal internal and external time step Dt ¼ 0:02s. The initial velocity was set zero and the free surface slope was gðx; y; 0Þ ¼ 0:1  0:01x  0:01y. The comparison of the computed surface elevation and analytical solution at the point x ¼ 0:5 m, y ¼ 5 m is shown in Fig. 1. The results indicate that the present model describes quite well this fully non-hydrostatic flow. 4.2 Steep wave transformation We investigated the steep wave transformation over the longshore bar. An experiment of Beji and Battjes (1993) was simulated. The flume geometry is shown in Fig. 2. A sinusoidal wave of 0:5 Hz frequency and 0:01 m amplitude is generated at the left open boundary. The computational domain was discretized on a grid with resolution 800 · 8 · 40. An equal internal and external time step Dt ¼ 0:01s was chosen for this example. The calculated water surface elevation g at 13:5 m and 19 m from the left open boundary is compared to experimental data in Figs. 3–4. As seen in the figures, the numerical model accurately predicts process of the development of the secondary non-linear-dispersive waves over a submerged bar.

4.3 Impact of solitary surface wave on vertical wall Here, results of a computation are compared with the laboratory experiments of Maxworthy (1976) and Zagriadskaya and Ivanova (1980) on the impact of the solitary surface wave of amplitude a on a vertical wall in the wave flume of length Lx ¼ 12:8 m, width Ly ¼ 0:77 m and depth H ¼ 0:32 m. The initial wave profile and velocity were described as: "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 3a x  x0 2 gð0; xÞ ¼ a cosh ð36Þ 4ð1 þ aÞ H g pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gH ð1 þ aÞ ; ð37Þ H where a ¼ a=H and x0 ¼ 0:3L m. As seen in Fig. 5, the model predicted well the maximum of elevation g at the wall. The resolution is 200 · 10 · 30. The equal internal and external time steps Dt ¼ 0:03s were used. The time dependence of wave pressure P on p the ffiffiffiffiffiffiffiffiffiundisturbed surface is shown in Fig. 6. Here T ¼ H=g. The nonhydrostatic model reproduced quite well, as observed in experiments, the characteristic two-peak profile of the pressure with depression near to the moment of maximum runup that was caused by the vertical acceleration of fluid at the wall. uð0; xÞ ¼

4.4 Mixing in lock-exchange flows The gravity currents are produced when fluid of a given density is released into fluid of different density. Experiments show (e.g. Hacker et al. 1996) that these flows are highly unstable and result in mixing between the fluid of the current and that of the ambient layer.

Fig. 3 Surface elevation at 13.5 m from the open boundary in the experiment (Beji and Battjes 1994) Fig. 1 Comparison of computations and analytical solution for surface elevation at the point x = 0.5 m, y = 5 m

Fig. 2 Bottom relief in the experiment (Beji and Battjes 1994)

Fig. 4 Surface elevation at 19 m from the open boundary in the experiment (Beji and Battjes 1994)

182

Fig. 7 Computed density distribution with dimensionless variables in the lock-exchange flow at time t=T ¼ 10

Fig. 5 Runup on the vertical wall as a function of soliton height. Comparison is given between experimental data of Maxworthy (1976) (1), Zagriadskaya and Ivanova (1980) (2) and calculations (3)

Fig. 8 Intrusion length as a function of time in experiment 302 Maderich et al. 2001 (1); calculated (2)

strates the ability of the present model to reproduce quite well the shear instability in stratified flows that is far from hydrostatic equilibrium. 4.5 Propagation of intrusive ‘‘bulge’’ in the thin pycnocline Fig. 6 Comparison of computed (solid line) time chart of wave pressure for dimensionless soliton height a=H ¼ 0:5(1) and 0:7(2) with experiment of Zagriadskaya and Ivanova (1980) shown by dashed line

With the use of the non-hydrostatic model, the lockexchange flow was simulated in a rectangular channel with depth 2H =0.1 m, length Lx =0.8 m and width Ly ¼ 0:15 m. The channel was partitioned into two compartments with a vertical barrier placed at a distance x0 ¼ Lx =2 from the one end. The buoyancy is gDq=q ¼ 1:0 cm/s2 . The calculations were carried out on a grid with spatial resolution of 400 · 10 · 100, internal time step DtI ¼ 0:03 s and external time step DtE ¼ 0:001s. This configuration of the computational experiment was similar to the DNS simulation of Hartel et al. (2000), who used the spectral numerical model. In contrast to that work, we simulated internal gravity flow with free surface. Fig. 7 shows the formation of Kelvin– Helmholz instability that results in finite-amplitude billows which appear similar to the experiment of Hacker et al. (1996) and the DNS simulations Hartel et al. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi (2000). Here, T ¼ qH =gDq. This example demon-

We investigated the formation and evolution of a large amplitude intrusive ‘‘bulge’’ that was generated by the collapse of the mixed region in the thin pycnocline. The results of simulation are compared with experiment 302 of Maderich et al. (2001). In this experiment a long tank was filled with two homogeneous layers of different density. The circular mixed region with average density of two layers collapsed in the centre of tank. The mixed region transformed into two ‘‘bulges’’ that moved in opposite directions. Owing to the symmetry of problem only half the tank was considered. The computational domain 2.1  0.4  0.19 m is discretized with resolution 250  10  60. Initial diameter of the mixed region was 0.19 m and buoyancy was gDq=q ¼ 4:2 cm2 =s. The internal time step was taken to be DtI ¼ 0:09 s and the ratio internal to external time step was 30. The resulting solitary bulge moved with almost constant speed (Fig. 8) that exceeded the values predicted by the theory of weakly non-linear long internal waves. As follows from Figs. 9 and 10 the model reproduces the form of the ‘‘bulge’’ quite well. One would expect that the model can be applied to the problems of the steep non-linear internal waves and intrusive flows.

183

Fig. 9 Computed intrusion visualized with markers

Fig. 10 Dye-visualized intrusion in experiment 302 (Maderich et al. 2001)

4.6 Exchange flows through long straits with a sill Here, the exchange flow through a shallow and narrow strait connecting two deep and wide basins filled with water of different density is considered. Such a flow is reproduced by the laboratory experiment 701 (Maderich 2000), which was then numerically simulated. The laboratory strait is 60.5 cm long with two deeper, broad basins and with a rectangular cross-section of maximal depth H = 8 cm and constant width Ly ¼ 0:9 cm (Fig. 11). The strait has a sill at its centre with a height h ¼ 4 cm. The buoyancy was gDq=q ¼ 1:48 cm=s2 . The density difference was maintained by heating the left basin and cooling the right. It was simulated by including source and sink terms SST ¼ ðT  T Þ=t in the heat-transport equation. In the left and right basins T are T0 þ 0:5DT and T0  0:5DT , respectively. Here T0 is the average temperature and DT is temperature difference, t is the relaxation time. The computations were carried out on a grid with resolution 200 · 11 · 60 in the curvilinear coordinates (Fig. 11). The internal step was 0.02 s and the ratio internal to external time step was 25. The flow is laminar; however, measurements and computations showed broadening of thermocline at the left side of the sill (Fig. 12, 13a). The detailed pattern of the velocity field shown in Fig. 13b indicates the presence of a complicated flow structure at the sill zone. The flow slows down at the left side of the sill, then creating a countercurrent. The dense water lifts along the left side of the sill and turns finally to left, forming a broad

Fig. 11 Curvilinear coordinate grid: horizontal view of channel

Fig. 12 Photo of experiment 701 (Maderich 2000)

interface layer. The broadening of the pycnocline downstream of sill or contraction was frequently observed in the sea straits; however, it was attributed to the turbulent entrainment between the fluid layers. The comparison of non-hydrostatic and hydrostatic calculations was carried out with the same resolution. Figure 13c indicates the essential difference between non-hydrostatic and hydrostatic velocity fields. It is about 22% at the slope and the foot of the sill. The terrain-following ocean models have difficulties in simulating baroclinic flows over steep topography because of the pressure gradient errors (Haney 1991; Mellor et al. 1998; Ezer et al. 2002). These errors are expected to overcome topographically generated nonhydrostatic effects on flow in some cases. To estimate pressure gradient errors, we carried out a run with the same strait geometry and numerical configuration, but with horizontally homogeneous and vertically linear density distribution and without forcing. The density difference between bottom and surface was the same as in the basic experiment. The standard POM density Jacobian scheme was used in the pressure gradient algorithm. Two parameters describe the steepness of the bottom topography: ‘‘slope parameter’’ sH ¼ jDH j=2H and ‘‘hydrostatic consistency parameter’’ rH ¼ jrDH =H Drj. The maxima of sH and rH are 0.025 and 2.7, respectively. In such a ‘‘moderately steep case’’ (Ezer et al. 2002) the maximum velocity of erroneous currents was 0:0122 cms1 whereas the maximum velocity in exchange flow (Fig. 13b) was 1:914 cms1 . Thus the effect of pressure gradient error was about 0:6%. Most of the terrain-following models retain the nontransformed horizontal viscosity and diffusion terms following the arguments of Mellor and Blumberg (1985). We estimated also the error caused by non-transformed horizontal viscosity and diffusion terms in the non-hydrostatic model. Two runs were performed: (1) with all terms in Eqs. (10) and (2) with only the first term in Eq. (10). The maximum difference in velocity fields was about 10% at the left slope of the sill. Therefore, we

184 Fig. 13 Computed density and velocity fields. a Contour plot of computed density from experiment 701. b Interface position in experiment 701 (Maderich 2000) (1) and nonhydrostatic velocity field with square root scaling (2). c Absolute value of difference between hydrostatic and non-hydrostatic velocity fields

should conclude that in this example it is necessary to use transformed horizontal diffusion terms.

5 Conclusions A non-hydrostatic model for simulating of free-surface unsteady stratified flows in the coastal sea has been developed. The primary purpose of the development of a new numerical algorithm is the extension of the class of hydrostatic model that uses time-splitting technique and terrain-following vertical coordinates. A number of algorithmic ideas are combined to design a numerical algorithm which is fully compatible with these models. The ability of the model to reproduce the non-hydrostatic effects was shown on several well-documented examples of non-hydrostatic flows. These examples demonstrate the efficiency and potential of the present model for the simulation of a wide spectrum of stratified flows with free surface. Acknowledgements This study was partially supported by US Civil Research and Development Foundation (Contract UG2-2425-SE02) and INTAS Fellowship for Young Scientists N YSF 2002-127 (Yuliya Kanarska). This article benefited from comments and suggestions of anonymous referees.

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