Generation of long waves in ice-covered lakes by moving disturbances of atmospheric pressure

34

9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China

2010, 22(5), supplement :34-39 DOI: 10.1016/S1001-6058(09)60165-7

Generation of long waves in ice-covered lakes by moving disturbances of atmospheric pressure Izolda V. Sturova Lavrentyev Institute of Hydrodynamics of SB RAS Novosibirsk, Russia E-mail: [email protected] ABSTRACT: The effect of ice cover on the oscillations of water in a two-dimensional constant-depth basin within the linear theory of long waves is considered. The ice cover is treated as thin elastic plate in the presence of compressive force. The ice is fastened hardly to a shore. The eigenfrequencies and eigenfunctions of free oscillations (seishes) are obtained. The forced oscillations of fluid and ice cover under the action of a moving disturbance of atmospheric pressure are investigated. The input data are taken to have values typical of Lake Baikal. The time dependence of total mechanical energy of the fluid is calculated for different velocities of pressure motion. The variations in ice-bending stresses are determined and the causes of shore-ice breaking are explained. KEY WORDS: shallow-water theory; ice-covered lake; seiches; moving pressure disturbance; ice-bending stresses..

1 INTRODUCTION Many lakes in the Northern hemisphere are covered by ice during a considerable part of the year. For the example, the Karelian lakes are under ice for more then six month a year [1~3], while Lake Baikal is under ice for about five month a year [4]. However, the hydrodynamic processes occurring in ice-covered basins and the ice-cover deformations caused by these processes are poorly known.

rest in the undisturbed state. In the linear treatment, the behavior of the fluid is described by the system of equations

∂u 1 ∂p ∂w 1 ∂p ∂u ∂w =− , =− , + = 0, (1) ∂t ρ ∂x ∂t ρ ∂z ∂x ∂z where u and w are the horizontal and vertical velocity components, respectively; p is the fluid pressure. The system of coordinates is arranged so that the horizontal axis x is perpendicular to the basin boundaries and the axis z is directed vertically upward. The vertical boundaries of the basin correspond to the values x = ± L . At the bottom and lateral boundaries of the basin, the no-flow conditions are given: w = 0 ( z = − H ), u = 0 (| x |= L). (2) At its top, the fluid is covered by an ice layer of constant thickness h and density ρ 1 . It is assumed that there is not air gap between the ice-cover bottom and water. The ice is modeled with a thin elastic plate [5]. Oscillations of ice cover and fluid are generated by the atmospheric pressure. The 2-D equation of elastic deflections of ice cover has the form

∂ 2ζ ∂ 4ζ ∂ 2ζ + D + Q + gρζ − p 0 = − P ( x, t ), ∂t 2 ∂x 4 ∂x 2 (3)

This paper describes the effect of ice cover on the oscillations of fluid in a 2-D closed basin. The ice cover is treated as thin homogeneous elastic plate in the presence of compressive force.

M

2 STATEMENT OF THE PROBLEM

where E is the Young’s modulus for the ice,

Let us consider a basin of breadth 2 L and constant depth H that is filled with an inviscid incompressible fluid with density ρ . The fluid is assumed to be at

M = ρ 1 h, D = Eh 3 /[12(1 − ν 2 )], Q = Kh

ν is

its Poisson’s ratio, ζ is the elevation of the ice/water interface; g is the gravitational acceleration; and

p 0 = p( x,0, t ) is the hydrodynamic water pressure

9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China at the lower surface of ice. P( x, t ) is a given function describing the external pressure on ice.

(in a generalized sense) system: 1

1

∫ U M [U ]dx = ∫ U N [U ]dx = 0 ( j ≠ n), j

Eq. (3) is a dynamic boundary condition at z = 0 for the system of Eqs. (1). The kinematic boundary condition has the form ∂ζ / ∂t = w at z = 0. At the ice edges, we consider rigid coupling of ice with the shore (fast ice) ζ = ∂ζ / ∂x = 0 (| x |= L). (4) Assuming that the depth of the fluid in the basin is much smaller that its horizontal size, we use the theory of shallow water [6], according to which the system of Eqs. (1) is substantially simplified ∂ζ 1 ∂ ⎛ ∂ 4 ζ ∂ 2ζ ∂ 2ζ ⎞ ∂u 1 ∂P ⎜⎜ D 4 + M 2 + Q 2 ⎟⎟ = − +g + , ∂t ∂x ρ ∂x ⎝ ∂x ρ ∂x ∂t ∂x ⎠ ∂ζ ∂u ∂t

+H

∂x

= 0,

(5)

with boundary conditions (2) and (4). At the initial time, the fluid and ice are at rest and no atmospheric disturbance is present: u ( x,0) = ζ ( x,0) = 0. (6) Then, we turn to the non-dimensional variables, using

L as the unit of length and

L / g as the unit of

time. Below, the following coefficients are used: D M Q δ= , γ = , χ= . 4 gρ L ρL ρ gL2

non-dimensional

3 FREE OSCILLATIONS First, let us investigate free long waves when no external disturbance pressure is present; therefore P ( x, t ) = 0 . We seek time-periodic solutions of the system of Eqs. (5) in the form u ( x, t ) = U ( x) sin ωt , ζ ( x, t ) = η ( x) cos ωt. (7) Substituting these relations into Eqs. (5) and boundary conditions (2) and (4), we obtain the ordinary differential equation for U (x) :

δU VI + χU IV + (1 − γω 2 )U ′′ + ω 2U / H = 0

(8)

with the boundary conditions

U = U ′ = U ′′ = 0

(| x |= 1).

(9)

The prime denotes differentiation with respect to x. Eq. (8) with boundary conditions (9) is an eigenvalue problem. It is convenient to write Eq. (8) in the form

M [U ] = λN [U ],

where

λ =ω

2

, M and N are the linear operators

M [U ] ≡ δU VI + χU IV + U ′′,

N [U ] ≡ γU ′′ − U / H .

Because this is a self-adjoint problem [7], all the eigenvalues are positive: λ n > 0 ( n = 1,2,...) . The eigenfunctions constitute a complete and orthogonal

35

n

−1

j

(10)

n

−1

where U j (x) and U n (x) denote the eigenfunctions corresponding to the eigenvalues λ j and

λ n . The

solution of Eq. (8) can be found in the form 6

U ( x) = ∑ c n exp k n x,

(11)

n =1

where k n are determined from the solution of the equation

δk 6 + χk 4 + (1 − γω 2 )k 2 + ω 2 / H = 0. Under certain limitations on the value

χ

, this

equation has two pure imaginary roots k1, 2 = ±iμ , and four complex conjugate roots k 3, 4 = α ± iβ ,

k 5, 6 = −α ± iβ , where μ , α , β > 0. Now, it is convenient to separate the eigenfunctions into symmetric and antisymmetric functions. For the s symmetric functions U (x) , Eq. (11) can be represented in the form

U s ( x) = b1 cos μx + b2 cosh k 3 x + b3 cosh k 4 x, where b1 , b2 , b3 are unknown constants. The use of boundary conditions (9) leads to the system of thirdorder linear equations for the unknown coefficients. If the determinant of this system is equal to zero, we obtain the following transcendental equation for the s eigenfrequencies ω n :

Im{k 3 [| k 3 | 2 tanh k 4 +

(12)

μ (k 3 tan μ − μ tanh k 3 )]} = 0.

Let us represent the corresponding eigenfunctions in the form:

U s ( x) = cos μx + 2 Re(b2 cosh k 3 x), b2 =

μ sin μ + k 4 tanh k 4 cos μ cosh k 3 (k 3 tanh k 3 − k 4 tanh k 4 )

.

The antisymmetric eigenfunctions U a ( x ) are found in the form

U a ( x) = c1 sin μx + c 2 sinh k 3 x + c3 sinh k 4 x, where c1 , c 2 , c 3 are unknown constants. We have the following equation for the eigenfrequencies

ω na :

Im{k 3 [tan μ (| k 3 | 2 tanh k 3 −

μ 2 tan k 4 ) − μ | tan k 3 | 2 ]} = 0. The corresponding eigenfunctions have the form

(13)

9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China

36

U a ( x) = sin μx + 2 Re(c 2 sinh k 3 x), c2 =

5 FREE SURFACE

μ cos μ tanh k 4 − k 4 sin μ cosh k 3 (k 4 tanh k 3 − k 3 tanh k 4 )

.

After the eigenfrequencies and eigenfunctions of the considered problem have been found, one can calculate the behavior of fluid and ice cover under the action of different types of external perturbations. 4 ACTION OF THE TIME-DEPENDENT EXTERNAL PRESSURE In the non-dimensional variables, the system of equations of motion (5) is written as

∂u ∂u ∂ζ ∂ ⎛ ∂4ζ ∂2ζ ∂2ζ ⎞ ∂P ∂ζ +H = 0 + + ⎜⎜δ 4 +γ 2 + χ 2 ⎟⎟ = − , ∂x ∂t ∂x ∂x⎝ ∂x ∂t ∂x ⎠ ∂x ∂t (14) with boundary conditions similar to (2), (4) and initial conditions (6). The solution of this system is sought as the expansion in terms of the symmetric and antisymmetric eigenfunctions:

′

∞

′

ζ ( x, t ) = ∑ [ An (t )U ns ( x) + Bn (t )U na ( x)],

(15)

n =1

u ( x, t ) = −

1 H

∞

∑ [ A (t )U n

s n

( x) + B n (t )U na ( x)] (16)

n =1

with the initial conditions

An (0) = A n (0) = Bn (0) = B n (0) = 0.

(17)

Here, the overdot denotes the time derivative and An (t ), Bn (t ) are the unknown functions to be found. Inserting these expansions into Eqs. (14), multiplying s a theirs sequentially by U m (x) and U m (x) , integrating with respect to from -1 to 1, and taking into account (10), we obtain the following simple differential equations for each of the unknown functions Am (t ), Bm (t ) ( m = 1,2,...)

The solution of the formulated problem is substantially simplified in the absence of ice cover. In this case, the first equation in (14) has the form

∂u ∂ζ ∂P , + =− ∂t ∂x ∂x

(21)

and the second equation remains unchanged. The boundary conditions have the form u = 0 at | x |= 1, and the initial conditions are similar to (6). The eigenfunctions of this problem are

U ns ( x) = cos μ ns x,

U na ( x) = sin μ na x,

μ ns = ω ns / H , μ na = ω na / H ,

ω ns = π H (n − 1 / 2), ω na = π H n.

(22)

Representing the solution of time-dependent problem (21) in the form of (15) and (16), we also obtain the final solution in the form (19), (20), but now we have

Λsn = Λan = − H −1 . For the free surface, it is easy to define the eigenfrequencies in the case of uneven bottom of the basin, that is, the depth of fluid is H (x) . Then the second equation in (14) has the form

∂ζ ∂ + (Hu) = 0. ∂t ∂x

In going to the non-dimensional variables, it is conveniently used the width of basin l = 2 L as the unit of length and

l / g as the unity of time. The

origin of new coordinate system is at the left boundary of the basin.

The free oscillations of the water in the form (7) are satisfied ( Hη ′)′ + ω 2η = 0 η ′ = 0 ( x = 0, x = 1). (23) s s s a a a  + λ A = Y (t ) / Λ , B  + λ B = Y (t ) / Λ , A This eigenvalue problem is named as Sturm – m m m m m m m m m m Liouville problem and usually solved by the shooting (18) 1 1 method. In this paper, the spectral method is proposed. Λ ms = ∫ U ms N [U ms ]dx, Yms (t ) = ∫ P( x, t )U ms ′dx, The function η (x) is sought in the form of an −1 −1 expansion 1 1 ∞ a a′ a a a 1 Λ m = ∫ U m N [U m ]dx. Ym (t ) = ∫ P( x, t )U m dx, η ( x ) = a0 + ∑ an cos nπx (0 ≤ x ≤ 1). −1 −1 2 n =1 The solution of Eqs. (18) with initial conditions (17) We substitute this expansion into Eq (23), multiply the has the form obtained relation by cos mπx ( m = 0,1,2,..) , and t 1 s s integrate it over x from 0 to 1. As a result, we obtain Am (t ) = s s ∫ Ym (ξ ) sin[ωm (t − ξ )]dξ , (19) ωm Λ m 0 a 0 = 0 and the set of linear algebraic equations t

Bm (t ) =

1 Y a (ξ ) sin[ωma (t − ξ )]dξ . a a ∫ m ωm Λ m 0

(20)

9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China ∞

∑a F n

nm

37

− ω 2 am = 0 ( m = 1,2,...),

n =1

1

Fnm = 2π2 nm ∫ H ( x ) sin nπx sin mπxdx. 0

Using the reduction method, the values ω may be determined as the eigenvalues of the square symmetric matrix Fnm by the standard numerical code. The spectral method can be applied also to the solution of the time-dependent problem. 2

6 NUMERICAL RESULTS

Fig. 1 Distribution of the depth

The physical parameters of the ice take the constant 3 values E =5GPa, ρ1 =860 kg/m and ν =0.3, with the

Table 1 Periods of seiche oscillations

density of the water ρ = 1008 kg/m . At first, the comparison of eigenfrequencies is made. The distribution of water depth H (x) is given in Fig. 1, which corresponds the fourth cross section of lake Vendyurskoe (the southern part of he Republic of Karelia, Russia) [1, 3]. The dashed line corresponds to the mean value of water depth H m = 6.96 m, the 3

width of basin is equal to l = 1760 m. The oscillation periods Tn = 2π / ωn for the first five modes are given FS

in Table, where Tn are the values for uneven bottom and the free surface determined by spectral M method; Tn and the results obtained from (22) for the rectangular basin with the constant depth H m , so M

called Merian’s formula Tn IC n

T

= 2l / gH m ; TnI and

are the results obtained from the solution of Eqs.

(12), (13) at the ice thickness h =0.5m, K = 0 and K 6

2

=2*10 N/m , respectively. As seen from Table, all oscillation periods for the given number n are very appeared to be closely allied. This means that the ice cover has almost no effect on the periods of seiche oscillations of lower modes. This fact was noted earlier in [1-3]. Merian’s formula gives a close approximation for the seiche periods. As stated in [1], the fact that the theoretically determined seiche period is close to the observed ones indicates that the ice cover does not have any apparent influence on the seiches. In the presence of ice cover, the eigenfunctions are also close to the eigenfunctions of the problem with a free surface. The differences are significant only in a vicinity of the shore. A time-dependent external influence is investigated for an example of a baric disturbance moving with a constant velocity.

Tn (min)

n

TnFS

TnM

TnI

TnIC

1 2 2 2 5

6.59 3.69

7.10

3.55

7.00 3.50

3.52

2.56 1.93 1.56

2.37 1.77 1.42

2.33 1.75 1.40

7.04 2.35 1.76 1.41

The input data are taken to have values typical of Lake Baikal: L = 25 km, H = 500 m. The compressive force takes no account of consideration. The function P( x, t ) in Eq. (3) is given in the form

P ( x, t ) = P0 F ( x, t ) , where ⎡π ⎤ F ( x, t ) = cos ⎢ ( x − xc (t ))⎥ (| x − xc (t ) |≤ R), ⎣ 2R ⎦ F ( x, t ) = 0 (| x − xc (t ) |> R ), P0 is a maximum deviation of pressure from the background value, 2R is the width of the disturbance area, xc (t ) = ct − R − L is the coordinate of the pressure epicenter, and c is the velocity of motion of the baric disturbance. At the initial time, the pressure epicenter is located outside the basin at the point x = −( R + L) , and the total time needed for the

pressure to pass over the basin is t1 = 2( R + L) / c. The calculations presented in Figs. 2, 4 and 5 are performed for P0 = 988.95 Pa ( a = P0 / ρ g = 10 cm), R = 10 km, and c = 2, 10 and 20 m/s. This range of velocities is substantially smaller than the critical velocity of long gravity waves for the given basin, which is equal to ccr = gH ≈ 70 m/sec. The calculations presented below performed with 15 eigenfunctions (symmetric and antisymmetric) in expansions (15) and (16). A further increase in the number of eigenfunctions has almost no effect on the result. The behavior of normal deflections of the ice cover at different times is given in Fig. 2. The deflections of

38

9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China

the ice cover at t / t1 = 0.2 and 0.4 for c = 2 m/sec and 10 m/sec are the same with a graphical accuracy. The ice deflections outside the coastal area 100 m in extent are almost independent of the ice thickness and repeat with a high accuracy the elevations of the free surface that arise from the action of this baric disturbance. After the disturbance crosses the right boundary of the basin (t > t1 ) , the fluid practically stops oscillating in the case of relatively slow disturbances with c = 2 m/sec and 10 m/sec. However, at c = 20 m/sec, where appear significant free oscillations close to a single-node seiche with the period T1 = 4 L /

gH ≈ 0.4 h. When the velocity

of motion of the baric disturbance is substantially smaller than c cr , free oscillations of the fluid appear in the basin as early as the external pressure moves over the basin. Depending on their phases, the free oscillations can both strengthen and reduce the effect of external pressure.

W (t ) . This can be done most simply in the case of a free surface. For linear long waves, we have 1 L W (t ) = ρ ∫ [ Hu 2 ( x, t ) + gζ 2 ( x, t )]dx. 2 −L If t ≥ t1 , the value of the energy remains constant and equal W (t1 ) , because the considered problem does not include dissipative processes. For the non2 dimensional quantity W = W (t1 ) /( gρa L) , in view of (15) and (16), we obtain

W =

L 2H

∞

∑ [ A

2 n

(t1 ) + B n2 (t1 ) + (ω ns ) 2 An2 (t1 ) +

n =1

(ω ) Bn2 (t1 )]. a 2 n

The dependence of W on the velocity of motion of the external pressure is shown in Fig. 3 for two values of the half-width of the disturbance area. For slow ( c ≤ 10 m/sec) disturbances, the oscillations of the fluid in both cases have a very low energy and are not shown in Fig. 3. In the velocity range considered, the dependence of W on c is non-monotonic and the energy of oscillations decreases with increasing R .

Fig. 3 Energy of oscillations of fluid after the passage of the baric disturbance. Curves 1 and 2 correspond to the values R = 10 km and 15 km, respectively.

Fig. 2 Distribution of normal deflections of the ice cover along the basin at different times. Curves 1-3 correspond to the values c = 2, 10 and 20 m/sec, respectively. The shaded rectangle indicates the position of baric disturbances.

To estimate the disturbance of the fluid, it is convenient to consider its total mechanical energy

The effect of ice cover is substantial only near the shore. The typical behavior of ice-cover deflections near the shore is shown in Fig. 4 for different values of ice thickness at c = 2 m/sec and t / t1 = 0.2. It can be seen that near the shore, a small increase in ice deflections occurs. The value of a local extremum for ice deflections is almost independent of ice thickness; however, the position of this extremum shifts toward the shore with increasing ice thickness.

9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China

39

investigated variations in ice-bending stresses and explained the causes of shore-ice breaking.

Fig. 4 Behavior of ice deflections near the shore. Curves 1-3 correspond to the values h = 0.2, 0.5 and 1 m,

respectively.

Such sharp variations in ice-cover deflections near the shore can lead to the breaking of fast ice; therefore, it is of interest to estimate the bending strength σ x of ice, which is equal to

Eh ∂ 2ζ . σx = − 2(1 − ν 2 ) ∂x 2 The bending strength of freshwater ice has a large scatter [8]: | σ x | ~ 0.37 – 2.32 MPa, which corresponds to the non-dimensional relation

| σ x | / E ~ 7.7 *10 −5 − 4.6 *10 −4. The magnitudes of bending stresses are the highest at the edges of the cover, i.e. in places of its coupling with the shore. The dependences of the ratio σ x / E at the point x = L are presented in Fig. 5 for c = 20 m/sec and different ice thickness. As the ice thickness increases, the magnitudes of shear stresses decrease. It can be seen that the fast ice may be broken. Maximum magnitudes of stresses at the right and left edges of the ice cover coincide and occur at about the time when the pressure epicenter passes over the corresponding boundary of the basin. When the disturbance crosses the right boundary of the basin, the bending stresses represent temporally undamped oscillations, thus reflecting the emergence of a single-node seiche with period T1 in the basin. As the distance from the ice edge increases, the bending stresses decrease sharply. 7 CONCLUSIONS The investigations performed by us made it possible to estimate the role of ice cover in the evolution of free and forced oscillations of water in a closed basin. It was shown that, for real parameters of freshwater ice, its effect can be found only near the shore. We

Fig. 5 Time dependence of bending stresses of ice at the righthand boundary of the basin. Curves 1 – 3 correspond to the values h = 0.2, 0.5 and 1 m, respectively.

ACKNOWLEDGEMENTS This study was supported by Siberian Branch of the Russian Academy of Sciences (Integration project 23). REFERENCES [1] Malm J, Bengtsson L, Terzhevik A, et al. Field Study on Currents in a Shallow, Ice-Covered Lake. Limnol Oceanogr, 1998, 43 (7):1669-1679. [2] Malm J. Some Properties of Currents and Mixing in a Shallow Ice-Covered Lake. Water Resour Res, 1999, 35 (1): 221-232. [3] Petrov M P, Terzhevik A Yu, Zdorovennov R E, et al. Motion of Water in an Ice-Covered Shallow Lake. Water Resour, 2007, 34 (2): 113-122. [4] Shimaraev M N, Verbolov V I, Granin N G, et al. Physical Limnology of Lake Baikal. A review. Baikal Int. Center for Ecological Research, 1994. [5] Kheysin D Ye. Dynamics of the Ice Cover. Gidrometeoizdat, Leningrad, 1967 (in Russian). Technical Translation FSTC-HT-23-485-69, U.S. Army foreign Science and Technology Center. [6] Stoker J J. Water Waves: The Mathematical Theory with Applications. Interscience, 1957. [7] Collatz L. Eigenwertaufgaben mit technischen Anwendungen. Geest and Portig, Leipzig, 1963. [8] Bogorodsckii V V, Gavrilo V P. Ice: Physical Properties and Current Methods of Glaciology. Gidrometeoizda Leningrad, 1980 (in Russian).