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 lengthscales 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
+ Δt f v
n
and v
n+1
= v
n
− Δt f u
n
Locations of our flui parcel are predicted with:
x
n+1
= x
n
+ Δt u
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)
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 lengthscales 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
+ Δt f v
n
and v
n+1
= v
n
− Δt f u
n
Locations of our flui parcel are predicted with:
x
n+1
= x
n
+ Δt u
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)