ISSN 0001-4370, Oceanology, 2016, Vol. 56, No. 4, pp. 491–500. © Pleiades Publishing, Inc., 2016. Original Russian Text © N.E. Voltzinger, A.A. Androsov, 2016, published in Okeanologiya, 2016, Vol. 56, No. 4, pp. 537–546.
MARINE PHYSICS
Nonhydrostatic Tidal Dynamics in the Area of a Seamount N. E. Voltzingera and A. A. Androsova, b aInstitute
of Oceanology, St. Petersburg Branch, Russian Academy of Sciences, St. Petersburg, 199053 Russia Alfred Wegener Institute for Polar and Marine Research, Handelshafen str. 12, Bremerhaven, 27570 Germany e-mail:
[email protected]
b
Received March 5, 2014; in final form, February 3, 2015
Abstract—The nonhydrostatic boundary problem for an arbitrary three-dimensional domain with a seamount is considered. The problem is integrated into curvilinear boundary-fitted coordinates on a nonuniform grid. In order to identify nonhydrostatic effects the grid is condensed on the slopes of the seamount preserving a coarse resolution in the rest of the domain, where the problem is solved in the hydrostatic approximation. Calculation results for the nonhydrostatic tidal dynamics and hydrology of the Strait of Messina in the area of a seamount are presented. DOI: 10.1134/S0001437016030243
1. INTRODUCTION The theory of long-wavelength disturbances is based on the assumption that the pressure in the fluid is hydrostatic. Admissibility of the law of hydrostatics is related to the parameter ε 2 = H 2 L2 , where H is the characteristic depth and L is the characteristic length. When the three-dimensional Euler equations are expanded in powers of small ε, the first approximation leads to hydrostatic equations, i.e., shallow water equations, whereas the second approximation yields a dynamic correction to the hydrostatic pressure. These well-known statements hold true if the characteristic scales have inherent meaning. In the case of a distinct change in the bottom topography (e.g., a seamount), the characteristic scales lose their global nature: the depth in the seamount area changes sharply, and the length of an incoming long wave decreases. Under such conditions, the hydrostatic approximation becomes less accurate and, accordingly, simulation of processes in the seamount area may require more complete nonhydrostatic equations. As follows from equations of vertical motion along the vertical section with scale of horizontal velocity U and buoyancy frequency N 2 = − g ρ z ρ 0 , where g is gravitational acceleration, ρ is density, and ρ0 is its reference value, the vertical acceleration of fluid particles can be neglected and the pressure is hydrostatic if 2
U ! 1 or Γ = ε Ri ! 1, 2 2 LN
(1)
where Ri = N 2H 2 U 2 is the Richardson number [12]. Under the conditions, H = O (102 m), N 2 ≈ 10 −4 s–1,
and U ≈ 0.5 m/s, the hydrostatic pressure condition is violated at wavelengths on the order of several kilometers, which corresponds to a mountainous area. The simulation of processes and phenomena on a nonhydrostatic scale was performed based on numerical solution of boundary value problems for equations of a viscous incompressible fluid, i.e., Reynolds-averaged Navier–Stokes (RANS) equations. Geophysical hydrodynamics phenomena with a horizontal scale of tens of meters to several kilometers include convective motions, short surface waves, internal waves, baroclinic instability, etc. [7, 10–13, 15]. In [4, 5], nonhydrostatic simulation is applied to calculate the thermohaline dynamics in the Gulf of Finland and the Barents Sea. This simulation considers gravity currents over a sloping bottom [9, 14, 16]. In addition, these works use both analytical and numerical solutions to identify parameters of subsequent processes: run-off of a relatively heavy liquid, mixing on the idealized shelf slope, nonhydrostatic barotropic-baroclinic interaction in a mountainous area [3]. Our work has been performed in the framework of the above simulation, which was added by the computational simulation of the nonhydrostatic dynamics, transforming a long wave in the seamount area. The next section formulates the nonhydrostatic boundary value problem for an arbitrary three-dimensional domain. The equations of motion, continuity, temperature, and salinity are given in Cartesian coordinates; the boundary value problem is formulated in curvilinear boundary-fitted coordinates and the given domain is mapped onto a parallelepiped. Section 3 considers the numerical method of integration of the transformed problem, using the subdivision of the pressure gradient into hydrostatic and dynamic com-
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ponents. This method is based on the solution to the hydrostatic problem, which is described in detail in [1]. The structure of the method is as follows: determination of the dynamic pressure gradient when solving the Poisson equation using the Laplace–Beltrami operator at each time step, and, then, determination of the nonhydrostatic contravariant vector in the divergence-free velocity field based on the dynamic pressure gradients. In Section 4, the nonhydrostatic dynamics and hydrology in the Strait of Messina are simulated. The most important morphometric element of the latter is the seamount in the narrowest part of the strait. The solution is achieved using a nonuniform curvilinear grid via interaction of the hydrostatic problem throughout the region and nonhydrostatic problem in telescoped subdomain of the seamount. Transition to integration of the nonhydrostatic problem is accomplished by computation without the need of a splicing procedure, solving two problems. The results include local and integral estimates of nonhydrostatic effects, an explanation of their character, and the influence on the formation of the density field of intermediate waters in the Strait of Messina. The statements and results of the work are discussed in Section 5. 2. STATEMENT OF THE PROBLEM 2.1. Equations in Cartesian Coordinates Let the unperturbed water surface coincide with the horizontal plane X 0Y of the right-hand Cartesian coordinate system and the axis 0Z be directed vertically upward. In the domain QT = Q × [0,T ] , where Q is a three-dimensional domain bounded by a free water surface ζ ( x, y; t ) , the bottom h( x, y), and the side surface ∂ Q, Q = {x, y, z; x, y ⊂ Ω, −h ≤ z ≤ ζ} , 0 ≤ t ≤ T , we consider the equations of motion, continuity, temperature, salinity and marine water conditions:
d u + 1 ∇ p + 2Ω × u = g + ∇ 2 ( K ∇ 2u ) + ( υ u z ) , (2) z dt ρ div u = 0, d Θi ∂Θ i = ∂ υ Θi + ∇ 2 ( K Θ i ∇ 2Θ i ) , dt ∂z ∂z ρ ( x, y, z; t ) = ρ ( Θ i ) ,
(3)
φ is latitude; K , υ are the horizontal and vertical turbulent viscosity coefficients, K Θi , υ Θ i are the turbulent diffusion coefficients i = 1, 2, Θ1 is temperature, and Θ 2 is salinity. System of equations (2)–(5) with respect to unknowns u, p, ρ, Θ i should be supplemented with turbulent closure to find coefficients υ, υ Θ i , K , K Θi and the equation for determining level ζ. This component of the solution follows from the vertically averaged equations for horizontal motion and the continuity equation. Vertical motion equation (2) is transformed to
∂p ∂z = − gρ
(6)
and determines the hydrostatic pressure pH. Assuming that ρ = ρ 0 + ρ ' (ρ ' ! ρ 0) and integrating (6) along the vertical, we have ζ
∫
p z =ζ − pH = − g ρ 0 ( ζ − z ) − g ρ ' dz.
(7)
z
Let us take the pressure at sea level as p ζ = const. It follows from (7) that the hydrostatic pressure gradient is the sum of its barotropic and baroclinic gradients: ζ
∫
∇ 2 pH = g ρ 0∇ 2ζ + g ∇ 2 ρ ' dz.
(8)
z
In the vertical motion equation of system (2), we have p = pH + pD , where pD is a dynamic pressure. Taking into consideration (6) we have
1 p + g ≈ 1 1 − ρ' ρ p + g ≈ 1 p + g ρ ρ ( z 0) z z 0 ρ ρ0 ρ0 = 1 ∂ ( pH + p D ) + g ρ ρ 0 = 1 ∂ p D . ρ 0 ∂z ρ 0 ∂z Let us represent the motion equation (2) in the form ∂ u + ∇ Π = φ, Π = p ρ , p ρ , p ρ , ( D 0) ∂t (9) × u + ∇ 2 ( K ∇ 2u ) + ( υ u z ) . φ = − ( u ⋅ ∇ ) u − 2Ω z
(4) (5)
where d dt = ∂ ∂ t + u ⋅ ∇, u = (u,v, w ) is the velocity vector, ∇ = (∇ 2, ∂ ∂ z ) , ∇ 2 = ( ∂ ∂ x, ∂ ∂ y ) is the horizontal operator gradient, ρ is density, p is pressure, and g = ( 0,0, − g ) is the gravitational acceleration vector. The Coriolis acceleration components are as follows: × u = ( f Hw − f v, fu, − f Hu ) , f H = 2Ω cos φ is its 2Ω sin φ is the vertical horizontal component, f = 2Ω is the Earth’s angular velocity vector, component, Ω
2.2. Boundary Value Problem in Curvilinear Coordinates Let us transform the equations to curvilinear coordinates:
ξ = ξ ( x, y ) , η = η ( x, y ) , σ=H
−1
(10)
( z − ζ ) , t ' = t,
where H = h + ζ, with the Jacobian of transformation J −1 = ∂ ( ξ, η, σ ) ∂ ( x, y, z ) , 0 ≠ J −1 < ∞, J = J H , * J = ∂ ( x, y ) ∂ ( ξ, η) is the Jacobian of the plane. If we * OCEANOLOGY
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choose four pairs of opposite sites of the side surface, then the domain Q is mapped onto parallelepiped Q*. Let us suppose that the physical domain Q is a strait with impermeable coastal boundaries ∂ Q1, mapped on the impermeable faces of the parallelepiped ∂ Q * and 1
open boundaries ∂ Q2 mapped onto its open faces ∂ Q2*; the upper and lower horizontal edges of the parallelepiped Q* are represented by Ω* rectangles in the planes σ = − 1 and σ = 0, respectively. The method of solving the nonhydrostatic problem presented below is essentially based on the solution to the hydrostatic problem when the pressure gradient (8) in curvilinear coordinates is as follows:
∇ 2 pH = g ρ 0∇ 2ζ + g ∇ 2I , 0
I = H ρ ' d σ, ∇ 2 = ∇ ξ i ∂ i . ∂ξ
(11)
∫
The form of equations and additional conditions in coordinates (10) is given below; the transformation algorithm is presented in detail in [1]. Equations (9) have the following form in boundary-fitted coordinates (10):
∂ u + ∇ Π = φ, φ = ϕ , ϕ , ϕ ( u v w) ∂t ×u = −U i ∂ u ∂ ξ i − W ∂ u ∂σ − 2Ω +H
( υ u σ ) σ + J * ( KJ * g u ξ −1
ik
k
)
ξi
(12)
.
i, k = 1, 2 for summation over repeated indices; U = U , 1
2 U = V , ξ1 = ξ, ξ 2 = η; W = σ t + v ∇ 2σ + wσ z is the contravariant vertical velocity; g ik = e i e k are the metric tensor components.
The continuity equation can be represented as
∂ JU i + ∂ JWˆ = 0, ∂σ ∂ξ i
(13)
where Wˆ = W − σ t . The hydrostatic equations of horizontal motion are as follows:
∂v + 1 ∇ p = φ , φ = ϕ , ϕ . ( u v) 2 H v v ∂t ρ
(14)
∂Θ + U i ∂ Θ + W ∂ Θ = Dˆ Θ . ( ) ∂t ∂σ ∂ξ i No. 4
Let the impermeable side faces ∂ Q1* of parallelepiped Q* lie in planes ξ = const, and its open faces ∂ Q * lie in planes η = const. Then
2016
U
= 0.
∂ Q1*
(16)
At the run-out in the open boundaries ∂ Q2* , we use linear extrapolation of the normal derivative to the plane η = const: contravariant components V have the following form:
1 g 2 jV , j ξ 22 g
(17)
where j = 1, 2, 3. At the inflow, two velocity components are given, ∂ Q2*
= ψ1 ( ξ, σ; t ) , V
∂ Q2*
= ψ 2 ( ξ, σ; t ) ,
(18)
as well as its normal derivative for a stable horizontal viscosity, K ∂ V ∂ n . The boundary conditions on the vertical are as follows: no-slip boundary condition at the bottom for the height of roughness parameter ε* and dynamic condition on shear stress τ ρ 0 :
U
i σ =− 1+ ε*
= 0, υU
i
= H ∇ξ τ ρ 0 . i
σ =0
(19)
The vertical component W satisfies the following conditions: W −1 = W 0 = 0. For Eqs. (15) at the solid boundary, the following is taken:
g Θξ j 1j
∂ Q1*
= 0.
(20)
The condition at the outflow through the open boundary is the zero normal derivative,
g Θξ j 2j
The equation for the density constituents Θ
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2.3. Boundary and Initial Conditions
U
)
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(
2
i
(
⎡ ⎛ ⎞ Dˆ = K Θ J −1 × ⎢ ∂ ⎜ J g 11 ∂ ⎟ * ⎣∂ ξ ⎝ * ∂ ξ ⎠ ⎛ ⎞⎤ 22 33 + ∂ ⎜ J g ∂ ⎟⎥ + ∂ υ Θ g ∂ . * ∂η ⎝ ∂ η ⎠⎦ ∂ σ ∂σ Vertically averaged equations for determining the level are added to the dynamic problem (12), (13). Closure for finding υ, υ Θ uses the turbulent kinetic energy equation, similarity relations, and the expression for the turbulence scale [1].
∂ V ∂ n ∂ Q* =
Here, U = v ∇ ξ are contravariant horizontal velocities; ∇ ξ i = e i = ξ ix , ξ iy is the contravariant basis vector; i
For the diffusion operator, the equation has a simplified form:
2
σ
−2
493
∂ Q2*
= 0,
(21)
and at the inflow, the characteristics of the incoming water mass, (15)
Θ ∂Q2* = ψ 3 ( ξ, σ; t ) .
(22)
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VOLTZINGER, ANDROSOV
The boundary condition at the bottom, consistent with the Laplace–Beltrami operator, is a conormal boundary condition:
∂Θ ∂ n σ =−1 = K Θ g Θ ξ i + υ Θ g Θ σ. 3i
33
or in projections at pD ρ 0 = p :
(u
(23)
∂Θ1 ∂Θ 2 = Qˆ ρ 0c p and ν Θ 2 = W s S, ∂ n σ =0 ∂ n σ =0
where Q is the radiant heat flux, c p is the specific heat capacity of seawater, and W s is the relationship between precipitation and evaporation. The initial conditions are the divergence-free velocity vector u t =0 = u 0 and the values of the constituents Θ i
t =0
= Θ i0.
(
which is determined from the equations
( v* − v ) + 1 ∇ k
τ
ρ
( w* − w ) = φ*. k
* = φ*, v
2 pH
τ
w
(26)
In order to implement the hydrostatic boundary problem (the first of Eqs. (26), continuity equation (13), and equations (4), (5) with boundary conditions (16)– (23)), the operators are split into coordinate directions with the second order of accuracy and the time step value defined only by the advective mode and the TVD procedure, which controls the behavior of solutions in the area of its sharp gradients and the multigrid procedure of convergence acceleration [1]. The algorithm for solving the nonhydrostatic problem is one of the modifications of the projection method for solving the Navier-Stokes equations. Subtracting Eq. (26) from (12), we have
(u
k +1
− u*
τ
) + 1 ∇p ρ
k +1 D
=0
k +1
+ pσ η x = 0,
k +1 i i ξy
)+ p
k +1
−U*
)+ g
+ pσk +1η y = 0, k +1
1j
pξk j+1 + g 13 pσk +1 = 0,
2j
pξ j + g pσ
(
)+ g
(
)+ g
τ k +1 ˆ W − Wˆ *
(24)
(25)
)+ p
ξ τ k +1 w − w*
τ V k +1 − V *
Let us add the following equation to hydrostatic equations (14):
u Hk +1 = u*,
k +1 i ξx ξi
σ η z = 0. τ Let us multiply the first of these equations by ξ x , and the second by ξ y ; then, we add them. Let us now add the first equation multiplied by η x and the second equation, multiplied by η y . Finally, let us sum the first equation multiplied by σ x , the second one multiplied by σ y , and the third one multiplied by σ z . As a result, we have
(U
Let the solution to problem (14), (24) with a time step ( k + 1) τ, k = 0, 1,..., k = [T τ] be known for u Γ = (u,v, w ) , which is considered a predictor of difference equation (12):
)+ p
(
3. METHOD FOR SOLVING THE NONHYDROSTATIC PROBLEM
w t = φ w .
− u*
τ k +1 v − v*
For the free surface, we have the following conditions:
ν Θ1
k +1
τ
3j
k +1
23
k +1
k +1
k +1
pξ j + g pσ 33
= 0,
(28)
= 0.
Multiplying each of these equations by J and differentiating the first equation over ξ, the second over η, and the third over σ, then adding them, we obtain
( Jg
ij
k +1
pξ i
)
ξj
= ⎡⎣( JU *) ξ + ( JV *) η + ( JWˆ *) σ ⎤⎦ τ , (29)
where, according to (13), it is taken into account that in curvilinear coordinates div U k +1 = 0. Elliptic equation (29) in Cartesian coordinates is the Poisson equation for the Laplace operator. In curvilinear coordinates, such an operator is called the Laplace– Beltrami operator. The equation is solved by the iteration method combining at each iteration cycle sweep along the vertical with the upper relaxation in the plane ( ξ, η) ,under conventional vertical boundary conditions:
∂p = 0, p σ =0 = 0; ∂ n σ =−1
(30)
on side impenetrable faces ∂ Q1* of the parallelepiped Q*, the following condition is given: ∂p = 0, ∂ n ∂Q1*
(27)
(31)
which follows from the first equation (28) at U OCEANOLOGY
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NONHYDROSTATIC TIDAL DYNAMICS IN THE AREA OF A SEAMOUNT
(a)
TYRRHENIAN SEA
(b) Farro
Italy
200
Ganzirri
50
50
Punta Pezzo Villa c.d.
0 20
Strait of Messina
Messina
N
20 0
50
50
Reggio
50
40 200 6000 80 0
50
0 60
40 0 200
50
35° 20° E
15°
50
S
Malta
Calabria
40 0
IONIAN SEA
10°
50
Sicily
Calabria
Sicily
200
Sicily
40° N
TYRRHENIAN SEA
Tunisia
495
IONIAN SEA
Fig. 1. Geographical position of Strait of Messina. (a) Bathymetric map and grid area with horizontal 33 × 83 resolution; (b) calculation domain of nonhydrostatic domain is outlined by dot–dash line. Longitudinal dashed line indicates position of axial section.
At the inflow ( η = 0 ) through the open boundary ∂ Q * , we have the boundary condition 2
∂p = ψ ( ξ, σ, t ) , ∂ n η=0
(32)
where, according to the second equation of system (28),
(
)
ψ k +1 = − V k +1 − V * τ g 22 is assigned to the iterative process at each cycle with the previous iteration:
(
ψ k +1,s = − V k +1,s − V *
)
τ g 22 , s = 1, 2, 3....
The solution to the problem (29)–(32) is determined by the dynamic pressure at which the gradient (28) is the k +1 nonhydrostatic velocity vector U k +1 = (U ,V ,Wˆ ) ,
and then the Cartesian velocity u k +1 from the relations: u = J (U η y − V ξ y ) , v = J (V ξ x − U η x ) , * * w = H (Wˆ − uσ x − v σ y ) ; level ζ is determined by the solution to the hydrostatic boundary value problem for the vertically averaged equations [1]. OCEANOLOGY
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4. NONHYDROSTATIC DYNAMICS IN THE STRAIGHT OF MESSINA 4.1. General Information The Strait of Messina separates Calabria (Italy) from the island of Sicily and connects the Ionian Sea with the Tyrrhenian Sea (Fig. 1a). The Strait of Messina extends for a distance of ∼20 km; its morphometry is characterized by variability in the coastline and a sharp variation in depth. At its narrowest part, the strait has the following parameters: a cross-sectional area of approximately 0.3 km2; minimal width and depth: width decreases to 3 km at the lowest depth of ∼70 m. The depth on both slopes of the seamount decreases sharply, reaching 1000 m at the southern part. In the Ionian Sea, the steep Messina canyon passes into the abyssal plain structure southeast of Sicily. To the north of the top of the seamount, coastal slopes are sharply divided, the depth increases, and the structure of the Strait of Messina turns toward the Tyrrhenian Sea as a giant submarine cone. The geometry of the strait and its location are the reason for the complex pattern of intense barotropic– baroclinic interaction. The results of studying the dynamics and hydrology of the Strait of Messina for a period of half the 20th century are given in [8]. The
496
VOLTZINGER, ANDROSOV
results of simulating the processes associated with the dynamics of the tidal strait based on numerical integration of two-dimensional and three-dimensional boundary value problems in the hydrostatic approximation are given in [2, 6]. Comparison of results of numerical solutions with observation data for diagrams of the velocity field in the narrowest part of the strait and the tidal level oscillation data obtained at coastal stations shows that, in general, hydrostatic description of the dynamics of the strait is possible. Significant differences can be expected only in some specific moments of the tidal cycle. The nonhydrostatic variations in hydrophysical characteristics of the Strait of Messina are described below in detail.
h0 ≤ h ≤ h. Thus, criterion (1) on seamountslopes is * transformed to
4.2. Computational Parameters and Accuracy of Estimating the Solution to the Nonhydrostatic Problem
4.4. Calculation of the Maximum Differences in the Velocity Fields Obtained by Solving Hydrostatic and Nonhydrostatic Problems
In order to check the convergence of the numerical solution, the boundary problem was integrated at different grid resolutions on grids: 33 × 83 × 40 (40 units along the vertical), 33 × 165 × 40 (reduced by half in the longitudinal direction), and 33 × 165 × 60. Solutions obtained using the second and third grids are slightly different, indicating the sufficiency of vertical resolution. These results where obtained by solving the problem on the 33 × 165 × 40 grid with steps of Δ min ≈ 30 m, Δ max ≈ 340 m (Fig. 1b) and a time step of T = 60 s. On the open boundaries, tidal level oscillations with amplitude and wave phase M2 are given. The initial density field is taken as depending only on salinity with a distinct pycnocline of about 80 m thick and salinity changing vertically from 37.5 to 38.5‰. The highest computational expenditures for solving the nonhydrostatic problem were related to the estimation of dynamic pressure according to (29)– (32). The nondivergent velocity vector is determined by the dynamic pressure gradients. The velocity field difference at integration of elliptic problem with accuracy of О (10–6 m2/s2) and О (10–7 m2/s2) is small enough to confine to the lower level of accuracy. In connection with this, it should be noted that an increase in accuracy up to (10–7 m2/s2) requires additional 200 iterations at each time step. 4.3. Results Let us calculate the violation of hydrostatics criterion (1) in the mountainous area of the Strait of Messina, where ε is the slope ε1/2 = H L = tanα but not the ratio of the characteristic scales. The Γvalue increases by several orders of magnitude. Another reason for the increase in the Γvalue is that the global meaning of characteristic depth in the mountainous area makes no sense and should be replaced by local slope depth h ( x, y ) . Let h0 be the depth over the top * of the seamount and h be a depth at its base, then
2 γ = U2 2 tan 2α. N h * In this case, the elevated γ values could not be invoked as a justification to neglect vertical accelerations, which is the basis of the conclusion (1). Some of the most significant features of the hydrophysical characteristics of the Strait of Messina used in the nonhydrostatic model are evaluated and discussed below.
Let us consider the flow difference in cross sections of the strait adjacent to seamount slopes. The maximum flow difference, which is the solution of two problems, δ q = q Γ − q H Γ,
q = JHV = H (v x ξ − uyξ )
in a tidal M2 wave cycle on the northern slope of the seamount, is revealed at a time instant close to the middle of the cycle. Figure 2a shows a cross section of the Strait of Messina indicating two verticals (I, II) where the differences are maximum. Zones of different flow directions are formed across the strait owing to coastal countercurrents with symmetric arrangement of zones where nonhydrostatics prevails with respect to the axis of the cross section. Figure 2b shows diagrams of the Cartesian velocity components on bold vertical lines. The highest deviation from the hydrostatics in the surface layer is observed for the horizontal velocity components on line I and for the vertical components on line II, reaching 5 mm/s in the middle part at a depth of about 100 m These differences define the local zones of maximum heterogeneity and diversity of the velocity fields where the contribution from the dynamic pressure component is the most significant. On the southern slope of the seamount, we have another pattern of differences (Fig. 3) with symmetrical arrangement of zones of predominant hydrostatic flow with respect to the axis of the cross section. The maximum flow difference occurs at the end of the tidal wave cycle M2, and the main difference is manifested only on the longitudinal velocity component v on vertical section II, reaching 10 cm/s. The vertical velocity diagrams on each of the lines are different, with the highest deviation in the near-bottom layer. Let us also present the calculated values of the characteristics μ ( u ) = ( u Γ − u Η Γ ) u Γ , which allows us to estimate the relative contribution of the dynamic pressure component. OCEANOLOGY
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(а) I
0
0 20 40 60 80 100
II
–50
–200
600 200 –200 –600
8
–1400
–250
300
0
5 10 15 20 25 v, cm/s
0 100 200
0
5 10 15 20 25 u, cm/s
300
0 10 20 30 40 50 u, cm/s
0
0 20 40 60 80 100
–1000
II
200
Depth, m
Flow difference, m3/s
Depth, m
–150
0 20 40 60 80 100
1000
0
100
–8 –4 0 4 v, cm/s
1400
–100
(b)
I
497
100 200
–16 –12 –8 –4 w, mm/s
0
300
–16 –12 –8 –4 w, mm/s
0
Fig. 2. (a) Section across northern slope of seamount. Plus—zone of predominant hydrostatic flow; minus— zone of predominant nonhydrostatic flow; (I, II) vertical sections of maximum difference in solution of two problems. (b) Diagrams of constituents of vector u = (u, v, w) in selected verticals. Solid line—nonhydrostatic flow; dotted line—hydrostatic flow.
(a)
II
25 1800
Flux difference, m3/s
Depth, m
50 75
100 125
1200 600 0 –600
150 –1200
175
–1800
0 40 80 120 160 200
I
(b)
0
II
40 80 120
–40 –30–20 –10 0 –12 –8 –4 v, cm/s v, cm/s
0 40 80 120 160 200
Depth, m
I
0
0
0
40 80 120
–40 –30–20 –10 0 –40 –30–20 –10 0 u, cm/s u, cm/s
0 40 80 120 160 200
0 40 80 120
–16 –12–8 –4 0 4 –30 –20 –10 w, mm/s w, mm/s
0
Fig. 3. (a) Section across southern slope of seamount. Plus—zone of predominant hydrostatic flow; negative— zone of predominant nonhydrostatic flow; (I, II) verticals of maximum difference in solution of two problems. (b) Diagrams of components of vector u = (u, v, w) in selected vertical sections. Solid line—nonhydrostatic flow; dotted line—hydrostatic flow. OCEANOLOGY
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16
(a)
10
(b)
3.0
2.0
2.5 8 2.0 1.2 1.5
8 4
||δ(ρ')||С, kg/m3
6
||δ(v)||С, cm/s
||δ(u)||С, cm/s
12
||δ(w)||С, cm/s
1.6
0.8 1.0
4 0
T/4
T/2
3T/4
2 T
0.5
0
T/4
T/2
0.4 T
3T/4
Fig. 4. Line of norm of maximum deviations in tidal wave cycle M2. (a) Horizontal components of velocity vector; solid line is u component; dashed line is v component; (b) vertical component of velocity vector (solid line) and density deviation (dashed line).
The diagrams of these characteristics show an increase in most pronounced dynamic pressure values with depth along the vertical sections I, II. Thus, on the northern slope, μ (u ) changes in the range (–0.4; 0.2) along vertical section I, and in the range (0.02; 0.1) along vertical section II, indicating a relative increase in the hydrostatic contribution with depth typical of tides. However, the behavior of μ ( w ) along vertical I of the northern slope and along vertical II of the southern slope shows no typical pattern. In both cases, the vertical velocity and, thus, the dynamic pressure component increase with depth. The same abnormal behavior is typical of the characteristics at a depth of ~20 m on section I of the southern slope where this characteristic varies in the range (20; –30). This is evidence of sharp predominance of the nonhydrostatic transverse velocity component in the coastal countercurrent. 4.5. Evaluation of Deviations of Hydrophysical Characteristics from Their Hydrostatic Values in the Maximum and Mean-Square Norms Let δ ( χ ) = χ H Γ − χ Γ be the difference in the component of the solution to the two problems in entire range of the tidal M2 wave cycle in the norms, N*
12
L2
2
−3
δ (v )
L2
≈ 4 × 10
сm/s, δ ( w )
δ (ρ ' )
L2
≈ 10 −4 kg/m3, δ ( ζ )
L2
L2
≈ 8 × 10 −4 сm/s,
≈ 5 × 10 −3 сm.
4.6. Other Nonhydrostatic Effects
δ ( χ ) C = max δ ( χ ) , δ (χ )
the maximum norm С for χ = {u,v, w, ρ '} are shown in Fig. 4. Note that the maxima of the norm should lie close to specific points of the tidal cycle, not necessarily coinciding with them. The velocity deviation line has two distinct maxima, which lie in the range of maximum flows, accounting for 5-10% of their maximum value. The density field deviation has three maxima, which confine to the moment of the change of currents, due to the nonhydrostatic violation of their reversibility, and to the interval of maximum currents. The maximum norms of the level of values have the same deviation, reaching maximum values at the time of the first energy peak, when δ ( ζ ) C ≈ 1.2 сm. The qualitative character of the behavior of the differences in the characteristics of the two problems is preserved in the mean-square norm on the smoothed line of its values in the tidal cycle. For the maximum energy range we have: δ (u ) L ≈ 6 × 10 −3 сm/s,
⎧⎪ ⎫⎪ , =⎨1 [δ ( χ )]2 ⎬ ⎩⎪N * N * ⎭⎪
∑
for the total number of values N* of the difference δ ( χ ) in nodes of the grid area. Comparison of the results in
The influence of the nonhydrostatic flow on the density field is expressed by the characteristic δρ'. There is a distinct difference in the compared fields on the southern slope at the time of maximum southward currents, when nonhydrostatic field density of heavier ionic water has a relatively more smoothed configuration isopycnic lines compared to the hydrostatic field OCEANOLOGY
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of steep isopycnals. Isopycnic surfaces are concentrated in surface layers due to the nonhydrostatic vertical velocity distribution. The structure of isopycnals in the case of a nonhydrostatic flow leads to the conclusion that light water in the surface layer moves further to the north and the water above the seamount becomes more stratified. The nonhydrostatic affects least of the surface elevation. One of the indices of the representativeness of the model of the Strait of Messina is calculation of amphydromy in the narrowest part of the strait and evaluation of the calculated level using the data obtained at coastal stations. The average vector comparison error for all stations in the hydrostatic model was 1.52 cm [6]. In the case of a nonhydrostatic flow, such an assessment changes insignificantly due to the minimum tidal level oscillations in amphydromic zone. The effect of the horizontal component of the cos φ, is estimated from the Coriolis force, f H = 2Ω vertical motion equation in system (12). The difference in solving the nonhydrostatic problem for constant f H and f H = 0 is very large, especially on the northern slope, where the longitudinal and vertical velocity are about 10 cm/s and 3 cm/s, respectively. 5. CONCLUSIONS The formulation and numerical method for solving the boundary value problem for three-dimensional Reynolds-averaged Navier-Stokes equations are considered. The problem is formulated in boundary-fitted coordinates that represent an arbitrary physical domain mapped onto a computational parallelepiped. The main difficulty in solving nonstationary Navier-Stokes equations is that the velocity field must satisfy the continuity equation with high accuracy, which does not include the time derivative. In the projection method, this is achieved with a two-step procedure: determination of the main pressure component at the first stage, then estimation of an amendment to the pressure by solution of the boundary value problem for the three-dimensional Poisson equation at each time step, so that the solution satisfies the continuity equation. It is evident that this main pressure component in terms of the geophysical hydrodynamics is the hydrostatic pressure and the amendment is the dynamic pressure caused by heterogeneity of the velocity field. Thus, the solution too the boundary value problem for the hydrostatic (primitive) equations is a specific problem and the basis for subsequent solution of the complete nonhydrostatic problem, which requires the largest computational expenditures. Since the hydrostatic solution of the problem is quite simple and a well-developed part of the general algorithm, our work focuses on implementation of its nonhydrostatic module, i.e., determination of the dynamic pressure by OCEANOLOGY
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solving the Poisson equation with the Laplace–Beltrami operator. An essential feature of this numerical method is solution of the nonhydrostatic problem not in the entire original domain, but only in an area where one can expect a significant manifestation of the dynamic pressure component due to some physical reasons. The solution in the subdomains is realized on a irregular curvilinear grid with a shock-capturing algorithm, which does not require the splicing procedure when solving two problems. We believe that this approach is particularly useful and effective for solving boundary value problems in mountainous areas with dominant vertical movement. This situation occurs in the simulation of long-wave processes, for which the law of hydrostatics is fulfilled with a high level of accuracy. However, the characteristic geometric scales in a mountainous area become uncertain and the hydrostatics admissibility test becomes meaningless. The results of calculating the nonhydrostatic effects of the tidal dynamics in the Strait of Messina are given as a supplement. We propose a modified criterion for violations of the hydrostatics on the slopes of the seamount. The areas where hydrophysical characteristics deviate from their hydrostatic values, reaching 10% of their maximum value, are localized, and the statistical estimates of such deviations in the tidal wave cycle M2 in the maximum and mean-square norms are presented. In addition, the influence of the dynamic pressure on the structure of the density field isolines is discussed, and the influence of the horizontal component of the Coriolis force, indicating the need to take it into account when calculating the vertical flow structure, was estimated. Our results show that even when calculating longwave movements in an mountainous area, it is worthwhile to refrain from hydrostatic simulation, as is commonly accepted, in favor of a detailed description with determination of the solution in the subdomains with a well-pronounced nonhydrostatic flow. The shock-capturing algorithm in the entire integration area of the boundary value problem is considered the most convenient procedure.
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Translated by D. Voroschuk
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