.Ocean Modelling for Beginners Phần 4 pptx

44 3 Basics of Geophysical Fluid Dynamics Fig. 3.12 Appearance of straight path (white line) of an object (ball) for an observer on a rotating turntable. The inner end of the white line shows the starting position of the object. The SciLab script “Straight Path.sce” in the folder “Miscellaneous/Coriolis Force” of the CD-ROM produces an animation 3.12.2 The Centripetal Force and the Centrifugal Force Consider an object attached to the end of a rope and spinning around with a rotating turntable. In the f xed coordinate system, the object’s path is a circle (Fig. 3.13) and the force that deflect the object from a straight path is called the centripetal force. This force is directed toward the centre of rotation (parallel to the rope) and, hence, Fig. 3.13 Left panel In the f xed frame of reference, an object attached to a rope and rotating at the same rate as the turntable experiences only the centripetal force. Right panel: The object appears stationary in the rotating frame of reference, which implies a balance between the cen- tripetal force and the centrifugal force. The SciLab script “Centripetal Force.sce” in the folder “Miscellaneous/Coriolis Force” of the CD-ROM produces an animation 3.12 The Coriolis Force 45 operates perpendicular to the object’s direction of motion. The centripetal force (per unit mass), which is a true force, is given by: Centripetal force =−Ω 2 r (3.31) where r is the object’s distance from the centre of the turntable, and Ω = 2π/T is the rotation rate with T being the rotation period. Per definition the rotation rate Ω is positive for anticlockwise rotation and negative for clockwise rotation. When releasing the object, it will fl away on a straight path with reference to the f xed frame of reference. In the rotating frame of reference, on the other hand, the object remains at the same location and is therefore not moving at all. Consequently, the centripetal force must be balanced by another force of the same magnitude but acting in the opposite direction. This apparent force – the centrifugal force – is directed away from the centre of rotation. Accordingly, the centrifugal force is given by: Centrifugal force =+Ω 2 r (3.32) When releasing the object, an observer in the rotating frame of reference will see the object flyin away on a curved path – similar to that shown in Fig. 3.12. 3.12.3 Derivation of the Centripetal Force The speed of any object attached to the turntable is the distance travelled over a time span. Paths are circles with a circumference of 2πr, where r is the distance from the centre of rotation, and the time span to complete this circle is the rotation period. Accordingly, the speed of motion is given by: v = 2π T r = Ωr. (3.33) During rotation, the speed of parcels remains the same, but the direction of motion and thus the velocity changes (Fig. 3.14). The similar triangles in Fig. 3.14 give the relation δv/v = δL/r. Since δL is given by speed multiplied by time span, this relation can be rearranged to yield the centripetal force (per unit mass): dv dt =− v 2 r , (3.34) where the minus sign has been included since this force points toward the centre of rotation. Equation (3.31) follows, if we finall insert (3.33) into the latter equation. 46 3 Basics of Geophysical Fluid Dynamics Fig. 3.14 Changes of location and velocity of a parcel on a turntable 3.12.4 The Centrifugal Force in a Rotating Fluid Consider a circular tank fille with flui on a rotating turntable. Letting the tank rotate at a constant rate for a long time, all flui will eventually rotate at the same rate as the tank. In this steady-state situation, the flui surface attains a noneven shape, as sketched in Fig. 3.15. The fina shape of the flui surface is determined by a balance between the centrifugal force and a centripetal force, that, in our rotating fluid is provided by a horizontal pressure-gradient force provided by a slanting flui surface. This balance of forces reads: − g ∂η ∂r =−Ω 2 r (3.35) where r is the radial distance from the centre of the tank. The analytical solution of the latter equation is: η(r) = 1 2 Ω 2 g r 2 −η o , (3.36) Fig. 3.15 Sketch of the steady-state force balance between centrifugal force (CF) and pressure- gradient force (PGF) in a rotating f uid. The dashed line shows the equilibrium surface level for the nonrotating case 3.12 The Coriolis Force 47 where the constant η o can be determined from the requirement that the total volume of flui contained in the tank has to be conserved (if the tank is void of leaks). The tank’s rotation leads to a parabolic shape of the flui surface and it is essentially gravity (via the hydrostatic balance) that operates to balance out the centrifugal force. The latter balance is valid for all flui parcels in the tank. The pathways of flui parcels are circles in the fi ed frame of reference. The observer in the rotating system, however, will not spot any movement at all. 3.12.5 Motion in a Rotating Fluid as Seen in the Fixed Frame of Reference With reference to a f xed frame of reference, flui parcels in the rotating tank exclusively feel the centripetal force provided by the pressure-gradient force. In the absence of relative motion, flui parcels describe circular paths. How does the trajectory of a flui parcel look like, if we give it initially a push of a certain speed into a certain direction? The momentum equations governing this problem are given by: dU dt =−Ω 2 X and dV dt =−Ω 2 Y (3.37) where (X, Y) refers to a location and (U, V ) to a velocity in the f xed coordi- nate system. On the other hand, the location of our flui parcel simply changes according to: dX dt = U and dY dt = V (3.38) Owing to rotation, velocities in the fi ed and rotating reference systems are not the same. Instead of this, it can be shown that they are related according to: U = u −Ω y and V = v +Ω x (3.39) where (x, y) refers to a location and (u,v) to a velocity in the rotating coordinate system. 3.12.6 Parcel Trajectory Before reviewing the analytical solution, we employ a numerical code (see below) to predict the pathway of a flui parcel in a rotating flui tank as appearing in the fi ed frame of reference. To this end, we consider a flui tank, 20 km in diameter, rotating at a rate of Ω = −0.727×10 −5 s −1 , which corresponds to clockwise rotation with a period of 24 h. 48 3 Basics of Geophysical Fluid Dynamics At location X = x = 0 and Y = y = 5 km, a disturbance is introduced such that the flui parcel obtains a relative speed of u o = 0.5 m/s and v o = 0.5 m/s. In the f xed coordinate frame, the initial velocity is U o = 0.864 m/s and V o =0.5m/s. The results show that the resultant path of the flui parcel is elliptical (Fig. 3.16). With a closer inspection of selected snapshots of the animation (Fig. 3.17), we can also see that the flui parcel comes closest to the rim of the tank twice during one full revolution of the flui tank. This finding which is simply the result of the elliptical path, is the important clue to understand why so-called inertial oscilla- tions, described below, have periods half that associated with the rotating coordinate system. Fig. 3.16 Trajectory of motion (white line) for one complete revolution of a clockwise rotating flui tank as seen in the f xed frame of reference. The SciLab script “Traject” in the folder “Mis- cellaneous/Coriolis Force” of the CD-ROM produces an animation Fig. 3.17 Same as Fig. 3.16, but shown for different time instances of the simulation. The tank rotates in a clockwise sense. The star denotes a f xed location at the rim of the rotation tank 3.12.7 Numerical Code In finite-di ference form, the momentum equations (3.37) can be written as: U n+1 = U n −Δt ·Ω 2 X n and V n+1 = V n −Δt ·Ω 2 Y n (3.40) 3.12 The Coriolis Force 49 where n is time level and Δt is time step. The trajectory of our flui parcel can be predicted with: X n+1 = X n +Δt ·U n+1 and Y n+1 = Y n +Δt · V n+1 (3.41) Again, predictions from the momentum equations are inserted into the latter equations as to yield an update of the locations. I decided to tackle this problem entirely with SciLab without writing a FORTRAN simulation code. 3.12.8 Analytical Solution Equations (3.37) and (3.38) can be combined to yield: d 2 X dt 2 =−Ω 2 X and d 2 Y dt 2 =−Ω 2 Y (3.42) The solution of these equations that satisfie initial conditions in terms of location and velocity are given by: X(t) = X o cos(Ωt) + U o Ω sin(Ωt) (3.43) Y (t) = Y o cos(Ωt) + V o Ω sin(Ωt) (3.44) This solution describes the trajectory of a parcel along an elliptical path. In the absence of an initial disturbance (u = 0 and v = 0), and using (3.39), the latter equations turn into: X(t) = X o cos(Ωt) −Y o sin(Ωt) Y (t) = Y o cos(Ωt) + X o sin(Ωt) which is the trajectory along a circle of radius  X 2 o +Y 2 o , as expected. 3.12.9 The Coriolis Force We can now reveal the Coriolis force by translating the trajectory seen in the f xed frame of reference (see Fig. 3.16), described by (3.43) and (3.44), into coordinates of the rotating frame of reference. The corresponding transformation reads: x = X cos(Ωt) +Y sin(Ωt) (3.45) y = Y cos(Ωt) − X sin(Ωt). (3.46) Figure 3.18 shows the resultant fl w path as seen by an observer in the rotating frame of reference. Interestingly, the flui parcel follows a circular path and completes the 50 3 Basics of Geophysical Fluid Dynamics Fig. 3.18 Pathway of an object that experiences the Coriolis force in a clockwise rotating f uid. The SciLab script “Coriolis Force Revealed.sce” in the folder “Miscellaneous/Coriolis Force” of the CD-ROM produces an animation circle twice while the tank revolves only once about its centre. Accordingly, the period of this so-called inertial oscillation is 0.5 T , known as inertial period, with T being the rotation period of the flui tank. Rather than working in a fi ed coordinate system, it is more convenient to for- mulate the Coriolis force from the viewpoint of an observer in the rotating frame of reference. In the absence of other forces, it can be shown that inertial oscillations are governed by the momentum equations: ∂u ∂t =+2Ω v and ∂v ∂t =−2Ω u (3.47) The Coriolis force acts perpendicular to the direction of motion and the factor of 2 reflect the fact that inertial oscillations have a period half that of the rotating frame of reference. If a parcel is pushed with an initial speed of u o into a certain direction, it can also be shown that its resultant path is a circle of radius u o /(2 | Ω) | ). With an initial speed of about 0.7 m/s and | Ω | = 0.727×10 −5 s −1 , as in the above example, this inertial radius is about 4.8 km. 3.13 The Coriolis Force on Earth 3.13.1 The Local Vertical In rotating fluid at rest, the centrifugal force is compensated by pressure-gradient forces associated with slight modificatio of the shape of the flui surface. On the rotating Earth, this leads to a minor variation of the gravity force by less than 0.4%. The local vertical at any geographical location is now define as the coordinate axis aligned at right angle to the equilibrium sea surface. This implies that, for a 3.13 The Coriolis Force on Earth 51 Fig. 3.19 Balances of forces on a rotating Earth fully covered with seawater in a state at rest. The gravity force (GF) is directed toward the Earth’s centre. The centrifugal force (CF) acts perpendic- ular to the rotation axis. The pressure-gradient force (PGF) balances the combined effects of GF and CF. The local vertical is parallel to PGF state at rest, the pressure-gradient force along this local vertical perfectly balances the combined effect of the gravity force and the centrifugal force (Fig. 3.19). 3.13.2 The Coriolis Parameter Owing to a discrepancy between the orientations of the rotation axis of Earth and the local vertical, the magnitude of the Coriolis force becomes dependent on geo- graphical latitude and Eqs. (3.47) turn into: ∂u ∂t =+f v and ∂v ∂t =−fu (3.48) where f = 2Ω sin(ϕ), with ϕ being geographical latitude, is called the Coriolis parameter. The Coriolis parameter changes sign between the northern and southern hemisphere and vanishes at the equator. This variation of the Coriolis parameter can be explained by a modificatio of the centripetal force in dependence of the orienta- tion of the local vertical (Fig. 3.20). Consequently, the period of inertial oscillations is T = 2π/ | f | and it depends exclusively on geographical latitude. It is 12 hours at the poles and goes to infinit near the equator. The radius of inertial circles is given by u o / | f | . Inertial oscillations attain a clockwise sense of rotation in the northern hemisphere and describe counterclockwise paths in the southern hemisphere. 52 3 Basics of Geophysical Fluid Dynamics Fig. 3.20 The centripetal force for a variation of the orientation of local vertical 3.13.3 The f -Plane Approximation The curvature of the Earth’s surface can be ignored on spatial scales of 100 km, to first-orde approximation. Hence, on this scale, we can place our Cartesian coordi- nate system somewhere at the sea surface with the z-axis pointing into the direction of the local vertical and use a constant Coriolis parameter (Fig. 3.21). The constant value of f is define with respect to the point-of-origin of our coordinate system. This configuratio is called the f-plane approximation. 3.13.4 The Beta-Plane Approximation The curved nature of the sea surface can still be ignored on spatial scales of up to a 1000 km (spans about 10 ◦ in latitude), if the Coriolis parameter is described by a constant value plus a linear change according to: f = f o +β y (3.49) Fig. 3.21 The sketch gives an example of a f-plane. The Coriolis parameter is given by f = 2Ω sin(ϕ), where ϕ is geographical latitude of the centre of the plane 3.14 Exercise 4: The Coriolis Force in Action 53 In this approximation, β is the meridional variation of the Coriolis parameter with a value of β = 2.2 × 10 −11 m −1 s −1 at mid-latitudes, and y is the distance in metres with respect to the centre of the Cartesian coordinates system definin f o . Note that y becomes negative for locations south of this centre. Equation (3.49) is known as the beta-plane approximation. A spherical coordinate system is required to study dynamical processes of length- scales greater than 1000 km. A discussion of such processes, however, is beyond the scope of this book. 3.14 Exercise 4: The Coriolis Force in Action 3.14.1 Aim The aim of this exercise is to predict the pathway of a non-buoyant flui parcel in a rotating flui subject to the Coriolis force. 3.14.2 First Attempt With the settings detailed in Sect. 3.12.6, we can now try to simulate the Coriolis force in a rotating flui by formulating (3.48) in finite-di ference form as: u n+1 = u n +Δtfv n and v n+1 = v n −Δtfu n Locations of our flui parcel are predicted with: x n+1 = x n +Δtu n+1 , and y n+1 = y n +Δt v n+1 The result of this scheme is disappointing and, instead of the expected circular path, shows a spiralling trajectory (Fig. 3.22). Obviously, there is something wrong here. The problem here is that the velocity change vector is perpendicular to the actual velocity at any time instance, so that the parcel ends up outside the inertial circle (Fig. 3.23). This error grows with each time step of the simulation and the speed of the parcel increases gradually over time, which is in conflic with the analytical solution. This explicit numerical scheme is therefore numerically unstable and must not be used. 3.14.3 Improved Scheme 1: the Semi-Implicit Approach Circular motion is achieved by formulating (3.48) in terms of a semi-implicit scheme: u n+1 = u n +0.5 α(v n +v n+1 ) and v n+1 = v n −0.5 α(u n +u n+1 ) (3.50) [...]... its local form is given by: ∂v ∂w ∂u + + =0 ∂x ∂y ∂z (3. 64) For a linear equation of state and the assumption that eddy diffusivities are the same for temperature and salinity, we can formulate an advection-diffusion equation for density anomalies, called the density conservation equation, that can be formulated as: ∂ρ + Adv(ρ ) = Diff(ρ ) ∂t (3.65) 3.17 Scaling 61 3.16.2 Boundary Conditions for Oceanic... The code also includes a formulation of the local-rotation approach with α ≈ Δt f that can be selected via the parameter “mode” 3. 14. 6 Sample Code and Animation Script The FORTRAN code for this exercise, called “Coriolis.f95”, and the SciLab script, “Coriolis.sce” can be found in the folder “Exercise 4 on the CD-ROM The fil “info.txt” contains additional information Fig 3. 24 Snapshots of the trajectory... intended 3. 14 Exercise 4: The Coriolis Force in Action 55 3. 14. 4 Improved Scheme 2: The Local-Rotation Approach The Coriolis force operates at a right angle to velocity and does not change the speed of motion, only the direction Hence, this feature can be simulated by a local rotation of the velocity vector; that is, u n+1 = cos(α)u n + sin(α)v n , v n+1 = cos(α)v n − sin(α)u n (3.53) (3. 54) From geometric... determined at α = 2 arcsin (0.5Δt f ) For Δt | f | . book. 3. 14 Exercise 4: The Coriolis Force in Action 3. 14. 1 Aim The aim of this exercise is to predict the pathway of a non-buoyant flui parcel in a rotating flui subject to the Coriolis force. 3. 14. 2. speed, which is certainly not intended. 3. 14 Exercise 4: The Coriolis Force in Action 55 3. 14. 4 Improved Scheme 2: The Local-Rotation Approach The Coriolis force operates at a right angle to velocity. expected. 3.12.9 The Coriolis Force We can now reveal the Coriolis force by translating the trajectory seen in the f xed frame of reference (see Fig. 3.16), described by (3 .43 ) and (3 .44 ), into coordinates of