Storyboard

The movement on the surface of the oceans arises from the interaction with the atmosphere and is conditioned by the deeper currents (more than 15 meters). In a first approximation, it can be considered as a flow at a constant velocity with stable vortices or those dragged by it.

>Model

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Iframe

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Video

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The movement of the ocean is generated on the surface by the movement of the air while in the depth by variations in density conditioned by temperature and salinity. Different effects are shown in the following NASA video:

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One of the most important currents in the Atlantic Ocean is the so-called Gulf Stream. It carries warm waters from the Caribbean to Europe, contributing to a milder climate in this area:

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Image

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The Gulf Stream originates from the Caribbean where there is also a series of circulation associated with the movements of the air masses in the region:

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In 1969 explorer Jacques Piccard's X-15 Ben Franklin submersible was swept away by the Gulf Stream. I float for this at a depth that corresponded to neutral flotation (between 180 to 610 m) and covered 2324 km:

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Concept

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To study ocean currents in the upper layer by measuring position (and with it speed), radiation, temperature and salinity, free buoys are used, which are called langrangian drifters or drifters:

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Concept

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There are different programs that have distributed drifers over all the oceans to monitor the flow in the ocean. An example is the Global Drifter Program (GDP) that presents the following distribution:

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Concept

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The rotational movement can be expressed as displacement in the x and y directions with values of the object distance from vortex center ($r$) and the object angle in the vortex ($\theta_w$), respectively. With coordinates the position x of the vortex center ($X$) and the position y of the vortex center ($Y$), we obtain that the position x of the object ($x$) is:

and for the position y of the object ($y$):

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Image

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The rotational movement can be expressed as displacement in the x and y directions with velocities of coordenada x de la velocidad del drifter ($u$) and coordenada y de la velocidad del drifter ($v$), respectively. With coordinates the speed x of the vortex center ($U$) and the speed y of the vortex center ($V$), we obtain that coordenada x de la velocidad del drifter ($u$) is:

and for coordenada y de la velocidad del drifter ($v$):

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Parameters

$\omega$

omega

Angular velocity of the object in the vortex

rad/s

$u$

u

Coordenada x de la velocidad del drifter

m/s

$v$

v

Coordenada y de la velocidad del drifter

m/s

$v_t$

v_t

Drifter tangential speed

m/s

$X_0$

X_0

Initial position x

m

$Y_0$

Y_0

Initial position y

m

$\theta_0$

theta_0

Object initial angle in the vortex

rad

Variables

$\theta_w$

theta_w

Object angle in the vortex

rad

$r$

r

Object distance from vortex center

m

$x$

x

Position x of the object

m

$X$

X

Position x of the vortex center

m

$y$

y

Position y of the object

m

$Y$

Y

Position y of the vortex center

m

$U$

U

Speed x of the vortex center

m/s

$V$

V

Speed y of the vortex center

m/s

$t$

t

Time from start of trace

s

Calculations

• Alternative method
• Traditional method
• Strategy
• 2D
• 3D

Equation

Number

First, select the equation: to , then, select the variable: to

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Calculations

Symbol

Equation

Solved

Translated

Calculations

Symbol

Equation

Solved

Translated

Variable Given Calculate Target : Equation To be used

Equations

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Equation

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The vortex moves in the $x$ direction with a constant of ($$), from ($$) reaching the time from start of trace ($t$) in $x$ The position x of the vortex center ($X$):

$X = X_0 + U t$

Conditions

Variables

$X_0$

Initial position x

$m$

8514

$X$

Position x of the vortex center

$m$

8506

$U$

Speed x of the vortex center

$m/s$

8510

$t$

Time from start of trace

$s$

8520

Relation

ID:(11495, 0)

Equation

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The vortex moves in the $y$ direction with a constant of ($$), from ($$) reaching the time from start of trace ($t$) in $y$ The position y of the vortex center ($Y$):

$Y = Y_0 + V t$

Conditions

Variables

$Y_0$

Initial position y

$m$

8515

$Y$

Position y of the vortex center

$m$

8507

$V$

Speed y of the vortex center

$m/s$

8511

$t$

Time from start of trace

$s$

8520

Relation

ID:(11496, 0)

Equation

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The vortex rotates steadily at ($$), starting from ($$) and reaching the time from start of trace ($t$) at ($$):

$\theta_w = \theta_0 + \omega t$

Conditions

Variables

$\omega$

Angular velocity of the object in the vortex

$rad/s$

8518

$\theta_w$

Object angle in the vortex

$rad$

8516

$\theta_0$

Object initial angle in the vortex

$rad$

8517

$t$

Time from start of trace

$s$

8520

Relation

ID:(11497, 0)

Equation

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The distance between the object at the position x of the object ($x$) and the position y of the object ($y$) and the center of the vortices at the position x of the vortex center ($X$) and the position y of the vortex center ($Y$) can be calculated using the Pythagorean theorem, resulting in the object distance from vortex center ($r$):

$r ^2=( X - x )^2 + ( Y - y )^2$

Conditions

Variables

$r$

Object distance from vortex center

$m$

8519

$x$

Position x of the object

$m$

8508

$X$

Position x of the vortex center

$m$

8506

$y$

Position y of the object

$m$

8509

$Y$

Position y of the vortex center

$m$

8507

Relation

ID:(11500, 0)

Equation

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If a body rotates at an angle of the object angle in the vortex ($\theta_w$) at a distance of the object distance from vortex center ($r$) from a center positioned at the position x of the vortex center ($X$), the result is ($$):

$x = X + r \cos \theta_w$

Conditions

Variables

$\theta_w$

Object angle in the vortex

$rad$

8516

$r$

Object distance from vortex center

$m$

8519

$x$

Position x of the object

$m$

8508

$X$

Position x of the vortex center

$m$

8506

Relation

ID:(11491, 0)

Equation

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If a body rotates at an angle of the object angle in the vortex ($\theta_w$) at a distance of the object distance from vortex center ($r$) from a center located at position the position y of the vortex center ($Y$), the result will be ($$):

$y = Y + r \sin \theta_w$

Conditions

Variables

$\theta_w$

Object angle in the vortex

$rad$

8516

$r$

Object distance from vortex center

$m$

8519

$y$

Position y of the object

$m$

8509

$Y$

Position y of the vortex center

$m$

8507

Relation

ID:(11492, 0)

Equation

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If we divide the relationship between the distance traveled in a time ($\Delta s$) and the radius ($r$) by the angle variation ($\Delta\theta$),

and then divide it by the time elapsed ($\Delta t$), we obtain the relationship that allows us to calculate the speed ($v$) along the orbit, known as the tangential velocity, which is associated with the angular Speed ($\omega$):

$v_t = r \omega$

$v = r \omega$

Conditions

Variables

$\omega$

$\omega$

Angular velocity of the object in the vortex

$rad/s$

8518

$r$

$r$

Object distance from vortex center

$m$

8519

$v$

$v_t$

Drifter tangential speed

$m/s$

10336

Relation

Development

As the mean Speed ($\bar{v}$) is with the distance traveled in a time ($\Delta s$) and the time elapsed ($\Delta t$), equal to

$\bar{v} \equiv\displaystyle\frac{ \Delta s }{ \Delta t }$

and with the distance traveled in a time ($\Delta s$) expressed as an arc of a circle, and the radius ($r$) and the angle variation ($\Delta\theta$) are

$\Delta s=r \Delta\theta$

and the definition of the mean angular velocity ($\bar{\omega}$) is

$\bar{\omega} \equiv\displaystyle\frac{ \Delta\theta }{ \Delta t }$

then,

$v=\displaystyle\frac{\Delta s}{\Delta t}=r\displaystyle\frac{\Delta\theta}{\Delta t}=r\omega$

Since the relationship is general, it can be applied for instantaneous values, resulting in

$v = r \omega$

.

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Equation

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Since the vortex rotates at the angular velocity of the object in the vortex ($\omega$) and is located ($$) from its center, the object moves at ($$):

If a body is at ($$) and the velocity in the $x$ direction is the speed x of the vortex center ($U$), then coordenada x de la velocidad del drifter ($u$) is:

$u = U - r \omega \sin \theta_w$

Conditions

Variables

$\omega$

Angular velocity of the object in the vortex

$rad/s$

8518

$u$

Coordenada x de la velocidad del drifter

$m/s$

9913

$\theta_w$

Object angle in the vortex

$rad$

8516

$r$

Object distance from vortex center

$m$

8519

$U$

Speed x of the vortex center

$m/s$

8510

Relation

ID:(11493, 0)

Equation

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Given that the vortex rotates at the angular velocity of the object in the vortex ($\omega$) and is located ($$) from its center, the object moves at ($$):

If a body is at ($$) and the velocity in the $y$ direction is the speed y of the vortex center ($V$), then coordenada y de la velocidad del drifter ($v$) is:

$v = V + r \omega \cos \theta_w$

Conditions

Variables

$\omega$

Angular velocity of the object in the vortex

$rad/s$

8518

$v$

Coordenada y de la velocidad del drifter

$m/s$

9914

$\theta_w$

Object angle in the vortex

$rad$

8516

$r$

Object distance from vortex center

$m$

8519

$V$

Speed y of the vortex center

$m/s$

8511

Relation

ID:(11494, 0)