Winds Generated by Flows and Riemannian Metrics
Savin Treanţă, Elena-Laura Dudaş
Faculty of Applied Sciences, University "Politehnica" of Bucharest, Bucharest, Romania
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To cite this article:
Savin Treanţă, Elena-Laura Dudaş. Winds Generated by Flows and Riemannian Metrics. American Journal of Science, Engineering and Technology. Vol. 2, No. 1, 2017, pp. 15-19. doi: 10.11648/j.ajset.20170201.13
Received: December 22, 2017; Accepted: January 9, 2017; Published: January 24, 2017
Abstract: The winds theory is based on PDEs whose unknown is the velocity vector field depending on time and spatial coordinates. The geometric dynamics is formulated using ODEs associated to a flow and a Riemannian metric, where the unknown is the velocity vector field depending on time. In this paper, we join these ideas showing that some geometric dynamics models generate winds. The second part of this paper is focused on the stability analysis of the considered models.
Keywords: Flow, Metric, Geometric Dynamics, Wind, Stability
1. Introduction
The wind usually refers to the horizontal component of the air motion relative to the earth. Mathematically, the wind is represented by the velocity vector field , where and . The forces applied to an element of air, moving almost horizontally over the earth’s surface with velocity , are:
(1) the inertial force, represented by the acceleration ;
(2) the pressure gradient force, represented by
(3) the deviating force due to the earth’s rotation, represented by ;
(4) the shearing stresses produced by the relative motion of the layers above and below, represented by .
Consequently, the partial differential equation of motion is of the form
(1)
and the winds are classified according to the relative importance of the four terms from the above relation:
Geostrophic wind: The motion is stationary (i.e. it is independent by ) and only terms and remain. The wind is expressed in terms of the horizontal gradient pressure and is along the isobars.
Gradient wind: The term is assumed to be approximately equal to , where is the radius of curvature of the isobars. The acceleration in the direction of motion is ignored. The wind is given in terms of the pressure gradient by a quadratic equation. The term is omitted.
Antitriptic wind: The terms and are dominant and represents the friction at the ground and it is therefore a force in the opposite direction to the motion. The wind is towards low pressure.
Ageostrophic wind: Departure from the geostrophic wind may be produced in a variety of ways in frictionless motion. The term is dominant and it expresses the convection at the earth’s surface.
The geometric dynamics (see [8, 9, 10]) is similar with the winds theory (see [6, 10, 4]): instead of the velocity vector field , where and , we work with the velocity vector field along a curve. In this case, the term doesn’t exist in the total acceleration, . For other connected viewpoints on this subject, the reader is directed to [1, 2, 3, 5, 7].
2. Winds and Geometric Dynamics
2.1. Hopf Geometric Dynamics and Hopf Wind
Consider the Riemannian manifold (see as the canonical (usual) metric in; = Kronecker's symbol) and the non-linear system of differential equations
(2)
which describes a bifurcation of Hopf type. Let be a vector field on , where
(3)
and let be the energy of the vector field . By a direct computation, we get and .
The Hopf geometric dynamics is described by
(4)
or
(5)
Now, let us explain how a wind is produced by the Hopf flow and by the Euclidean metric. Firstly, we extend the Hopf vector field from to . In this sense, we introduce
.
The vector field can be written as , where . Consequently, in vector notation, the wind produced by the Hopf flow and by the Euclidean metric is described by the second-order equation
(6)
2.2. Rabinovich Geometric Dynamics and Rabinovich Wind
Here, we use the Riemannian manifold and consider the following non-linear system of differential equations
(7)
which is known as Rabinovich-type system. Let be a vector field on , where
(8)
and let be the energy of the vector field By direct computation, we obtain and .
Remark 2.2.1 It is well-known that the divergence of a vector field defines the speed of contraction-dilation of the volumes by the flow generated by the vector field. In our case, we obtained , so is a solenoidal vector field, that is, the Rabinovich flow conserves the areas.
The Rabinovich geometric dynamics is described by
(9)
or
(10)
Let us introduce the wind produced by the Rabinovich flow and by the Euclidean metric. We remark that can be written as where So, in vector notation, the wind produced by the Rabinovich flow and by the Euclidean metric is given by the second-order equation
(11)
2.3. Van der Pol Geometric Dynamics and Van Der Pol Wind
Let us start with the Riemannian manifold and consider the following non-linear system of differential equations
(12)
which is known as van der Pol oscillator, being a control parameter. Let be a vector field on , with
(13)
and let be the energy of the vector field . By a simple calculation, we get and .
The van der Pol geometric dynamics is described by
(14)
or
(15)
The wind produced by the van der Pol flow and by the Euclidean metric is determined as follows. We extend the van der Pol vector field from to . In this direction, we introduce the vector field with . The vector field can be written using the Monge representation Thus, in vector notation, the wind produced by the van der Pol flow and by the Euclidean metric is given by the second-order equation
(16)
2.4. Phytoplankton Geometric Dynamics and Phytoplankton Wind
Further, we shall consider a biological model. We also use the Riemann manifold Define the non-linear system of differential equations
(17)
which describes the Phytoplankton Growth Model, where is the substrate, is the phytoplankton biomass and is the intracellular nutrient per biomass.
Let be a vector field on , where
(18)
and let be the energy of the vector field By simple computations, we get and
The Phytoplankton geometric dynamics is described by
(19)
or
(20)
Let us introduce the wind produced by the Phytoplankton flow and by the Euclidean metric. The vector field can be written using the Monge representation So, in vector notation, the wind produced by the Phytoplankton flow and by the Euclidean metric is described by the second-order equation
(21)
3. Stability Analysis of Considered Models
3.1. Stability Analysis of Hopf Bifurcation
The equilibrium point of the bifurcation of Hopf type is the solution of the following algebraic system
(22)
where is a parameter. In this case, we get . Denoting
(23)
the linearization around the equilibrium point is
(24)
that is,
(25)
where is defined as in the previous. By direct computation, we get (see as the Jacobian matrix of the function computed at ). Solving the equation , we obtain the solutions . Consequently, for the previous linearized system is unstable, for is asymptotically stable, and at (i.e. ) we find so the linearized system is stable (see as algebraic multiplicity, respectively geometric multiplicity of ).
Let us summarize the previous analysis as
Proposition 3.1.1 For the non-linear system of differential equations (bifurcation of Hopf type)
(26)
the linearization around the equilibrium point is
(27)
Moreover, the linearized system is: asymptotically stable ; stable ; unstable .
3.2. Stability Analysis of Rabinovich System
We shall follow the same steps as in the previous case. The algebraic system
(28)
gives us the equilibrium point of the Rabinovich system. The linearization around the equilibrium point is
(29)
that is,
(30)
By a direct computation, we get (see as the Jacobian matrix of the function , computed at Solving the equation , we obtain the multiple solution . Consequently, the linearized system is not asymptotically stable. By , we conclude the linearized system is stable.
Proposition 3.2.1 For the non-linear system of differential equations (Rabinovich system)
(31)
the linearization around the equilibrium point is
(32)
Moreover, the associated linearized system is stable but not asymptotically stable.
3.3. Stability Analysis of Van Der Pol System
Solving the following non-linear algebraic system
(33)
we get the equilibrium point of the van der Pol system. The associated linearized system (around the equilibrium point) is
(34)
that is,
(35)
By direct computation, we get (see as the Jacobian matrix of the function computed at ). We get . There are three cases:
a) So, for the associated linearized system is asymptotically stable, and for it is unstable.
b) . The solutions are given by . For the system is unstable (the both solutions do not have negative real part). For the system is asymptotically stable (the both solutions have negative real part).
c) The solutions are given by . For the linearized system is asymptotically stable. For the linearized system is unstable. Finally, for () we get that is the associated linearized system is stable.
Proposition 3.3.1 For the non-linear system of differential equations (van der Pol system)
(36)
the linearization around the equilibrium point is
(37)
Moreover, the linearized system is: asymptotically stable ; stable ; unstable.
3.4. Stability Analysis of the Phytoplankton Growth Model
Solving the following non-linear algebraic system
(38)
we get the equilibrium points , or , or of the Phytoplankton Growth Model. The associated linearized systems (around the equilibrium points) are
(39)
that is,
(40)
By a direct computation, we get (see A as the Jacobian matrix of the function computed at ). Solving the equation , we obtain the solutions , so the associated linearized system is unstable. For the equilibrium point we get (see B as the Jacobian matrix of the function computed at ). The equation gives the solutions Consequently, the associated linearized system is unstable. For the equilibrium point we get (see C as the Jacobian matrix of the function computed at ). The equation gives the solutions Consequently, the associated linearized system is asymptotically stable.
Proposition 3.4.1 For the non-linear system of differential equations (Phytoplankton Growth Model)
(41)
the linearizations around the equilibrium points
(42)
are
(43)
Moreover, the associated linearized systems are: unstable (in the case of the equilibrium points ) and asymptotically stable (in the case of the equilibrium point.
4. Conclusions
In this paper, we considered some special differential systems (taken from the literature) and we studied the associated geometric dynamics. Taking into account the winds theory, we succeeded to derive some second-order differential equations which permit us to describe the winds produced by the special flows (Hopf, Rabinovich, etc.) and Euclidean metric. Also, a stability analysis of considered models is provided.
Acknowledgements
We would like to thank Professor Constantin Udriste for bringing into our attention this subject and for valuable discussions during the preparation of our work.
References