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VIRTUAL GEOPHYSICAL FLUID DYNAMICS LABORATORY

G. SÓTI 1 and Á. BORDÁS 2

1Department of Physics, Faculty of Sciences, University of Novi Sad, Serbia soti@neobee.net

2University Centre for Meteorology and Environmental Modelling, University of Novi Sad, Serbia

abordas@uns.ns.ac.yu Introduction Java simulations are very useful in visualization and provide new opportunities in educational and scientific applications of the web. Virtual experiments in geophysical fluid dynamics were designed with an idea to link the theoretical lectures, problem solving sessions and laboratory demonstrations. Taking advantages of Java in visualization our simulations present analytical solutions and the time dependence of these solutions on the northern hemisphere. The simulations demonstrate effects of gravity as well as combined effects of rotation, gravity and stratification. This report represents an introduction and provides a short theoretical overview of the treated problems assuming basic knowledge in fluid dynamics.

The Virtual Geophysical Fluid Dynamics Laboratory is available on the http://soti.on.neobee.net/virtualgfdlab web site. For the source codes, please contact one of the authors. Simulations 1. Rotating Tank [further reading Cushman-Roisin (1994)] The problem of the cylindrical rotating tank is very popular in problem solving sessions. The rotation of the tank causes fluid’s depression in the middle and rising up along the rim of the tank. This continues until the resulting pressure gradient prevents any further displacement. In the equilibrium state, at any point on the free surface the downward gravitational force and the outward centrifugal force combine into a net force orthogonal to the free surface (see Figure 1-1.). The curvature of the free surface is actually neutralizing the rather small centrifugal force.

Figure 1-1. Equilibrium surface of a fluid in the rotating tank [from Cushman-Roisin (1994)]

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The problem of the rotating tank is axial symmetric, where the rotational axis is coinciding

with vertical axis. The equations of motion in horizontal and in vertical direction are respectively

01 2 =+∂∂

− rrp ω

ρ and 01

=−∂∂

− gzp

ρ, (1.1)

where r is the horizontal coordinate, z is the vertical coordinate, p is the pressure, ρ is the fluid density and ω denotes the angular velocity of rotation. For total differential of the pressure the using equations (1.1) we get

gdzrdrdzzpdr

rpdp ρρω −=

∂∂

+∂∂

= 2 . (1.2)

After integration equation (1.2) we can represent in form

gzrpp ρρω −=−2

22

0 (1.3)

Everywhere on the free surface the p = p0, thus the solution for the shape of free surface is

grz

2

22ω= . (1.4)

The angel of the slope α is defined by

g

rdrdztg

2ωα == . (1.5)

Java simulation of the rotating tank is shown on Figure 1-2. Fixed parameters are the radius of

the tank r = 5 cm, fluid height in the tank without rotation H = 5 cm and the gravitational acceleration g = 9.81 m/s2.

Figure 1-2. Rotating Tank – Java simulation

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2. Seiche [further reading Gill (1984)] Seiche is a standing wave formed in an enclosed or semienclosed body of water. The initial displacement of water from a level surface can arise from a variety of causes and the restoring force is gravitational force which always tends to maintain a level surface. The period of oscillations depends on geometry and dimensions of the tank or basin. The term of seiche was promoted by François – Alphonse Forel to describe the rhythmic oscillations of the water surface in Lake Geneva. The word originates in a Swiss French dialect word that means "to sway back and forth".

In the absence of bottom slope the governing equations are the linearized frictionless shallow water equations for a rectangular channel

x

gtu

∂∂

−=∂∂ η (2.1)

and

xuH

t ∂∂

−=∂∂η , (2.2)

where u denotes the horizontal velocity, H is the undisturbed uniform water depth, η is the small vertical displacement from the undisturbed water depth and g is the gravitational acceleration. From equations (2.1) and (2.2) we can easily get the standard wave equation

2

22

2

2

xc

t ∂∂

=∂∂ ηη , gHc = . (2.3)

The time period of seiche oscillation is

gHnLT 2

= , (2.4)

where L is the length of tank or basin and n is the order of oscillations.

Java simulation seiche is presented on Figure 2. Fixed parameter is the gravitational acceleration g = 9.81 m/s2.

Figure 2. Seiche – Java simulation

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3. The Dam-Breaking Problem [abridged version of Cushman-Roisin (2004)] The governing equations for the surface waves in the case of frictionless unsteady flow in the absence of a bottom slope for a rectangular channel are the shallow water equations

xhg

xuu

tu

∂∂

−=∂∂

+∂∂ (3.1)

and

xuh

xhu

th

∂∂

−=∂∂

+∂∂ , (3.2)

where u is the horizontal velocity, h is water depth and g is the gravitational acceleration.

The multiple quadratic terms render these equations strongly nonlinear, but some near symmetry between the two permits a transformation in some form that more easily leads to a solution. For this let us substitute the water depth by h the quantity c defined as

ghc = , gch

2

= . (3.3)

This new quantity c has the dimensions of speed, just like the remaining variable u. With this notation, the equations become:

xcc

xuu

tu

∂∂

−=∂∂

+∂∂ 2 (3.4)

and

xuc

xcu

tc

∂∂

−=∂∂

+∂∂

2 . (3.5)

The equations have quadratic nonlinearities in all terms that do not include the temporal derivative.

Adding twice (3.5) to (3.4) yields

( ) ( )cux

ccux

ut

22 +∂∂

−=+⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂ ,

or, after gathering all terms on the left:

( ) ( ) 02 =+⎥⎦⎤

⎢⎣⎡

∂∂

++∂∂ cu

xcu

t. (3.6)

Likewise, if we subtract twice (3.5) from (3.4), we get

( ) ( )cux

ccux

ut

22 −∂∂

=−⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂ ,

which in turn can be made more compact:

( ) ( ) 02 =−⎥⎦⎤

⎢⎣⎡

∂∂

−+∂∂ cu

xcu

t. (3.7)

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We recognize here a hyperbolic problem, which may be solved by the method of characteristics. Indeed, equations (3.6) and (3.7) state that the quantity )2( cu + remains unchanged when traveling at the speed )( cu + and that )2( cu − remains unchanged when traveling at the speed

)( cu − respectively. The quantities )2( cu ± are the Riemann invariants of the problem. From this, we can conclude that c acts as a wave speed that either adds to the flow velocity,

)( cu + , or subtracts from it, )( cu − . Thus, there is a single kind of wave, which is able to propagate information both with and against the flow. Since ghc = implicates gravity, this is naturally a gravity wave. [Important note: Although the equations are nonlinear, the present representation of the gravity wave does not include the fullest of possible nonlinearities. Indeed, the assumption of a vertically uniform velocity and the use of the hydrostatic balance for the determination of pressure does not permit the investigation of wave breaking.]

An interesting application of the preceding dynamics is the dam-breaking problem. For this, we imagine an initial condition consisting in water piled to a depth H behind a dam situated at x = 0 and dry land on the other side. The bottom is horizontal everywhere and there is no initial velocity (Figure 3-1a). Thus, the initial conditions are

Hh = , 0=u for 0<x , and 0=h , 0=u for 0>x . (3.8)

The dam suddenly fails at t = 0, and for the sake of simplicity, it is imagined that it is completely removed instantaneously. The problem is to describe the flooding on the formerly dry side and the water depletion on the backside. This can be done analytically by the method of characteristics.

The first thing to notice is that finite water depth and finite water speed can only generate finite values of )( cu ± . Therefore, the signals propagating upstream and downstream do so at finite speed and, at some finite time t, the situation consists in three regions, a far upstream region [ ])(tax −< and a far downstream region [ ])(tbx > , each still untouched and bracketing zone [ ])()( tbxta <<− where change is taking place (Figure 3-1b).

Figure 3-1. The dam-breaking problem: a) initial conditions and b) some time later [from Cushman-Roisin (2004)]

In the far upstream region, a point A at some early time (see Figure 3-2) is crossed by a pair of

characteristics that carry the Riemann invariants )2( cu ± , which are equal to gH2± . It follows that

gHcu 22 =+ and gHcu 22 −=−− . The solution is obviously still 0=u and ghc = .

Progressing through this region, we conclude that 0=u and ghc = through the region, as expected. The limiting characteristics is the one that begins at the dam site and travels backwards, namely

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gHcdtdx −=−=/ or gHtx −= . Along that characteristics, u and c remain constant 0 and gh , respectively, setting an initial or a boundary condition on the upstream side of the changing region.

Since the complementary quantity ( )cu 2+ is carried unchanged along characteristics such as P-R and Q-S emanating from this line, it follows that

gHcu 22 =+ throughout the region in which u motion occurs, and equation (3.6) is trivially satisfied.

Figure 3-2. Characteristic configuration for the dam-breaking problem [from Cushman-Roisin (2004)]

Let us now take two points, P and Q, along the aforementioned characteristic and follow their

respective progression along the other family of characteristics to the points R and S, respectively, as shown in Figure 3-2. Along the P-Q and R-S paths, the quantity ( )cu 2+ is conserved and thus

gHcucu 222 =+=+ PPRR

gHcucu 222 =+=+ QQSS which yields

2R

R

ugHc ==

2

SS

ugHc == .

If points R and S lie on the same crossing characteristics, they must share the same value of ( )cu 2− value and therefore RRSS cucu 22 −=− , from which follows that

⎟⎠⎞

⎜⎝⎛ −−=⎟

⎠

⎞⎜⎝

⎛ −−2

22

2 RR

SS

ugHu

ugHu

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and therefore RS uu = , RS cc = and RRSS cucu −=− .

From this follows that the crossing characteristic has a constant slope and is therefore a straight line. Since point R lies at an arbitrary distance from point P, we can further generalize by stating that all ( )cu − characteristics in the region of movement are straight lines, with

const.=−= cudtdx

Time integration provides ( ) 0)( xtcutx +−= . Since we know that gHcu 22 =+ everywhere in the

region, the characteristic slope cu − must be equal to cgH 32 − . But, the initial value of h is

undetermined at 0=x , and there are many ghc = possible values with which to begin characteristics. There thus exists a family of characteristics fanning outward from the same point (x = 0, t = 0), each with its own slope cgH 32 − , ranging between gH− (for h = H) to gH2 (for h = 0). This is also the only family of characteristics, for those that emanate from 0>x all begin with u = c = 0 and do not propagate. It follows that the zone of motion consists entirely in the set of straight characteristics that emanate from x = 0 with the aforementioned range of slope. The leading edges (Figure 3-1) are thus gHtta =)( and gHttb 2)( = .

For this set of characteristics starting from the origin (x0 = 0), the progression of x with time is given by ( )cgHtx 32 −= . Inverting this equation to solve for c tildes

t

xgHc33

2−= (3.9)

which is the solution of the problem in terms of c. Switching back to the water depth h, we have

2

22

994

94

gtx

gtgHxH

gch +−== , (3.10)

which describes an upward-facing parabola starting from h = 0 at the leading edge gHtx 2= , passing through h = 4H/9, a fixed value, at x = 0, where the dam used to be, and ending with h = H at the backward end gHtx −= (see Figure 3-1).

The velocity is obtained from the gHcu 22 =+ constraint:

txgHu

32

32

+= , (3.11)

which varies from zero at the backward end to gH2 at the leading edge. At the location of the

broken am (x = 0), the velocity u and wave speed c are both equal to ( ) gH3/2 . The discharge per unit width of channel is given by

3

3

2

2

272

92

278

gtx

gtgHxgHH

huq +−== , (3.12)

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which varies from zero at ax −= back to zero at bx = . It must thus reach a maximum in the region of motion. Taking the x-derivative and setting it to zero, we note that there are two extrema, the first at x = 0 (at the former location of the dam) and x = b (the forward edge). Since q(x = b) = 0, the maximum is the value at x = 0, namely

( ) gHHqq2780max == (3.13)

which is invariant over time.

Java simulation of the dam-breaking problem is shown on Figure 3-3. Fixed parameter is the gravitational acceleration g = 9.81 m/s2.

Figure 3-3. The Dam-Breaking Problem – Java simulation 4. Storm Surge [following Tél (2003) and Gill (1982)] Storm surge refers to an abnormally high sea level produced by extremely strong wind blowing constantly parallel to the coastline in the case when the Ekman drift [Gill (1982) and Cuchman-Roisin (1994)] is directed normal to the coast. The phenomena of storm surge we can easily describe considering a semi-infinite sea of uniform depth H bounded by straight boundaries (see Figure 4-1.). The constant wind stress is parallel to the coast and of magnitude τ . The problem is to find the motion and the changes of sea level that the wind causes.

Figure 4-1. Storm surge cased by extremely strong wind blowing constantly parallel to the coastline [from Tél (2003)]

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In absence of the longshore variation and friction, the governing equations are the wind forcing linearized shallow water equations

x

gfvtu

∂∂

−=−∂∂ η ,

Hygfu

tv

ρτη

−∂∂

=+∂∂ (4.1)

and

0=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

yv

xuH

tη , (4.2)

where u and v denotes the vertically averaged currents, H is the undisturbed water depth, η is the vertical displacement, f is the Corilolis parameter [ ϕsin2Ω=f , where Ω is the angular velocity of the Earth’s rotation and ϕ denotes the latitude] and ρ is the water density.

Satisfying the initial conditions of the rest and two boundary conditions [u vanishes at the coast (x = 0) and η vanishes far offshore ( +∞→x )] we get

( )RxeHf

u /1 −−−=ρτ , Rxe

Htv /−−=ρτ (4.3)

and

RxeRf

t /−=ρτη , (4.4)

where fgHR = is the Rossby radius of deformation. Because there is a steady flow toward the coast the sea level will change linearly in time. Current u is independent of time as well as current v , parallel to the coast, is always in geostrophic equilibrium with the pressure gradient normal to the coast.

Java simulation of storm surge is shown on Figure 4-2. Fixed parameters are the onshore wind stress τ = 2 N/m2, the angular velocity of the Earth’s rotation Ω = 7.29 10-5 1/s and the gravitational acceleration g = 9.81 m/s2.

Figure 4-2. Storm Surge – Java simulation

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5. Coastal Upwelling [following Tél (2003), Cushman-Roisin (1994) and Gill (1982)] One of the most important case of internal motion of stratified ocean is the coastal upwelling, in which the wind driven Ekman transport [Gill (1982) and Cuchman-Roisin (1994)] confined to the upper layer, is away from the coast. This requires deep water to upwell to replace the surface water being removed. Coastal upwelling can be described considering semi-infinite two layer model of sea by straight boundaries (see Figure 5-1).

Figure 5-1. Phenomena of coastal upwelling

In absence of the longshore variations and friction the equations of motion for two layer model assuming small deviations for upper and lower layer respectively are

x

gfvt

u∂∂

−=−∂∂ η

11 ,

11

1

Hfu

tv

ρτ

=+∂∂ , (5.1)

xhg

xgfv

tu

∂∂

+∂∂

−=−∂∂ '2

2 η , 022 =−

∂∂

fut

v, (5.2)

where χ−≈ 1Hh [ χ is a displacement of lower layer] denotes the thickness of upper layer,

( )gg 1/' ρρΔ= is the reduced gravitational acceleration, τ is the longshore wind stress and all other symbols are conventional used in previous section (Storm Surge). The equations of continuity for upper and lower layer are

xu

Hth

∂∂

−=∂∂ 1

1 , x

uH

th

∂∂

−=∂∂

− 22 . (5.3)

From equations (5.1), (5.2) and (5.3) using definitions 21 uuu −≡ and 21 vvv −≡ we will get

xhgfv

tu

∂∂

−=−∂∂ ' ,

1Hfu

tv

ρτ

=+∂∂ (5.4)

and

xu

HHth

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

∂∂

−1

21

11 . (5.5)

Solving our system of differential equations satisfying the initial conditions of rest and two boundary conditions [u vanishes at the coast (x = 0) and h = H1 far offshore ( +∞→x )] we will get

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( )'/

1

1 RxeHf

u −−=ρτ , '/

1

RxeH

tv −=ρτ (5.6)

and

'/

21

21 '

RxeHH

HfR

tHh −

+−=

ρτ , (5.7)

where f

HHHH

gR 21

21''

+= is the internal Rossby radius of deformation.

Apart from linear approximation, we shall consider the situation when the wind is sufficiently strong or is blowing far a sufficiently long time. In this case the lower and denser layer can rise to the surface forming a front (see Figure 5-2a). Continued wind action displaces the front to xc offshore distance and exposing colder water to the surface (see Figure 5-2b). This mature state is called full upwelling.

a) b)

Figure 5-2. Coastal upwelling: a) formation of a front and b) offshore front displacement [from Tél (2003)]

If we assume that the wind moves the surface layer as a solid slab we would get the

intermediate structure of full upwelling shown by dashed line on the Figure 5-2b, where xEk is the Ekman displacement [Gill (1984) and Cushman-Roisin (1994)]. Such a situation can not persist and an adjustment must follow, causing an onshore spread over a distance equal to the internal Rossby radius and we have a final structure shown on Figure 5-2b. by solid line.

Java simulation of coastal upwelling is presented on Figure 5-3. Fixed parameters are the onshore wind stress τ = 0.1 N/m2, the angular velocity of the Earth’s rotation Ω = 7.29 10-5 1/s and the gravitational acceleration g = 9.81 m/s2.

Figure 5-3. Coastal Upwelling – Java simulation

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References

Cuchman-Roisin B.: 1994, Introduction to Geophysical Fluid Dynamics, Prentice-Hall, New Jersey Cuchman-Roisin B.: 2004, Hydraulics of Rivers and Lakes, School and Workshop on Oceano-

graphy, Lakes and Rivers, CIM, Lisbon Gill A. E.:1982, Atmosphere-Ocean Dynamics, Academic Press, San Diego Tél T.: 2003, Környezeti áromlások - kézirat, ELTE, Budapest (in Hungarian)