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Chapter VII. Rotating Coordinate Systems

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Chapter VII. Rotating Coordinate Systems Chapter VII. Rotating Coordinate Systems 7.1. Frames of References In order to really look at particle dynamics in the context of the atmosphere, we must now deal with the fact that we live and observe the weather in a non-inertial reference frame. Specially, we will look at a rotating coordinate system and introduce the Coriolis and centrifugal force. Rotating and Non-rotating Frames of Reference First, start by recognizing that presence of 2 coordinate systems when dealing with problems related to the earth: a) one fixed to the earth that rotates and is thus accelerating (non-inertial), our real life frame of reference b) one fixed with respect to the remote "star", i.e., an inertial frame where the Newton's laws are valid. Apparent Force In order to apply Newton's laws in our earth reference frame, we must take into account the acceleration of our coordinate system (earth). This leads to the apparent forces that get added to Newton's 2nd law of motion: dVr Fr = real (inertial) (7.1) dtm r dVr Fr F =+real apparent (non-inertial) (7.2) dtmm Fr where apparent are due solely to the fact that we operate / observe in a non-inertial m reference frame. rr dAdA We need to relate to . dtdt inertialnon-inertial 7-1 7.2. Time Derivatives in Fixed and Rotating Coordinates Reading Section 7.1 of Symon. Using concepts developed in the last chapter, let's introduce some notion that will prove to be useful. r W z y Vr r T R r Latitude circle a =colatitude f = latitude x Equatorial Plane Let r = position vector from the origin (center of the earth). Then, we can find out that the instantaneous tangential velocity of a particle at any distance r ( azr + r where a = mean radius of earth and z = altitude above ground) is given by rrr VrT =W´ . (7.1) r Here W is the vector angular velocity of the earth rotation. r We can see this by first looking at the magnitude of VT . According to the above figure, rrrrrr |VT |=|W|R=|W||rr|sina =||W´ (7.2) 7-2 r dA Now, let's try to relate in 2 reference frames: dt a) Fixed (absolute) system (iˆˆ,,)jkˆ r b) Rotating ( W ) system (iˆˆ',jk',ˆ') z z' y' r W y x x' r Now, a vector A is the same vector no matter what coordinate system it is viewed from. This r ˆˆˆˆˆˆ A=Axi++AyjAzxk=A'i'++A'yzj'Ak''. r Given that the primed system is rotating, however, the time derivative of A will be different if viewed from the 2 systems. Mathematically, we have 'a' for absolute r dAdA dA dA ax=iˆˆ++y jkz ˆ dtdtdtdt dA' dA' dA' =x iˆˆ'''++y jkz ˆ (7.3) dtdtdt diˆˆ'dj''dkˆ +++AAA''' xdtyzdtdt Note that the primed unit vectors vary with time! (Remember our earlier derivation of the velocity in plane polar coordinates?) 7-3 iˆ'()tt+D Diˆ' due to rotation of unit vector itˆ'() rrr r Recall earlier that VrT =W´ for a particle rotating at angular velocity W . Since drr Vr = therefore T dt drr r = W´r (7.4) dt The above is correct only for a vector whose length does not change. The same is true for the unit vectors, therefore diˆ' =W´r iˆ' dt djˆ' =W´r ˆj ' (7.5) dt dkˆ' r =W´kˆ' dt r dA If we now loot at in the reference frame of the rotating (primed) system, then dt iˆˆ',jk' and ˆ' appear to be fixed! r dAdA' dA' dA' rz=x iˆˆ'''++y jkˆ (7.6) dtdtdtdt r which is a time rate of change in A seen from the rotating reference frame and is r dA analogous to a earlier. dt Thus, from (7.3), we can write 7-4 r r dA dA a = r +A'(Wr´iˆˆ')+A'(Wrr´j')+Ak'(W´ ˆ') dt dt xyy rr dAdA r = rr+Wrr´(A'iˆˆ'+A'j'+A'kAˆ') =+W´ à dtxyydt r r dA dA r a = =r +W´r A (7.7) dt dt r Let's look at a physical interpretation. If Ar= r , then we can write drr drr a =r +W´r r (7.8) dt dt Absolute Velocity relative Velocity of the velocity in to moving coordinate the inertial (rotating) system itself frame V coordinates ()V (remember a r rr Vr=W´r ) rrr In another word, Vabs=+VVrelcoord . Think of the earlier example of throwing a baseball towards a person on the ground from a moving railway car. If you can remember this simple analogy, you will be able to remember the formula in the box. 7-5 7.3 Equation of Motion in Absolute and Rotating Coordinates Let's now apply concepts from the previous section to any arbitrary velocity vector of a particle assuming that the origins of the rotating and fixed (inertial) frames are the same. In this case, the position vector to any location is the same in both systems. Our goal is to find the equations of motion in the absolute and rotating coordinates. Now, we just showed that rrrr Va =Vr+W´ (7.9) r where V is the relative velocity (we shall drop subscript r from now on). r Apply Eq.(7.7) to Va in (7.9) à dVr d a =éVr+Wr´rrrù+Wr´éùVrrr+W´ dtdt ëûëû dVdr =+éWr´rrrù+Wrr´Vr+Wrr´éùW´ dtdt ëûëû (7.10) dVrrdW drr =+´rrr+Wr´+Wrr´Vr+Wrr´éùW´ dtdtdt ëû dVr =+2Wrr´Vr+Wrr´éùW´r dt ëû Apply the Newton's second law à r r r Fi dV dV å ai=+2Wrr´Vr+Wrr´éùW´=r (7.11) dtdtmëû The part in the box is the equation of motion in the rotating coordinate system! It describes the change of (relative) velocity in time subjecting the net force. The forces on the right hand side are real forces, and the second and third term on the left arises because of the coordinate rotation, and there are apparent (not real) forces. We will discuss them in more details in the following. 7-6 We will look at the last term one the left with respect to an earth oriented coordinate system: r W z y r VT r 2 r R -W R r Latitude circle a =colatitude f = latitude x Equatorial Plane Note that |Rrr |=|r |cosf . Look at the last term in (7.11): rrr W´ëûéùW´r . rrrr 2 SHOW IT FOR YOUSELF that W´ëûéùW´rR=-W ! (7.12) (Hint – use the right hand rule to find out the direction of this vector, then use the definition of cross product to determine its magnitude) Using this we can rewrite equation (7.11) as r r r Fi dV dV å ai=+2Wr´VRrr-W=2 (7.13) dtdtm Abs. Accel Rel. Accel. Coriolis accel Centripetal accel Net real force This is a VERY IMPORTANT EQUATION. 7-7 How do we make use of this equation? Let's look at the forces on the right hand side of (7.13) in the real atmosphere. They are Gravity = gr (force per unit mass) 1 PGF (per unit mass) = -Ñp (you have seen this before and you will r derive this term in Dynamics I) Friction = …. (neglected for now) These are real forces (those seen and felt in an inertial / fixed reference frame), so let's put them into the equation: r dV rrr2 1 +2W´V-WR=-Ñgpr (7.14) dt r this is the acceleration that we see and measure in our earth coordinate system – not an inertial frame, but the next 2 terms take that into account. Rearranging (7.14) à Coriolis force Centrifugal force r dV 1 rrr2 =-Ñp+gr -2W´VR+W (7.15) dt r Relative Real forces Apparent forces due acceleration solely to the rotation of the coordinate system Equation (7.15) is "THE" EQUATION OF MOTION used in meteorology – it is the backbone of meteorology! 7-8 7.4 Forces in Equation of Motion We are now going to spend more time looking in details at each of these terms. 7.4.1. Gravity Let's look first at gravity. Recall that GMm grr = e ˆ (7.16) a where Me = mass of earth and a = mean radius of the earth. It turns out it is convenient to combine gRr and W2 r , the centrifugal force. Due to the presence of the centrifugal force that's directed away from the axis of rotation, the earth is actually elliptic, with the radius at the equator slightly larger than that at the poles. If one measures that force that's pulling a mass downward, it is the sum of true gravity (the one due to gravitational effect) and the centrifigual force, i.e., rr2 r gnet =gR+W . (7.17) r r Note that g points towards the center of the earth while gnet is slightly equatorward. r Since gnet is what we really measure at the surface of the earth, from now on we drop the subscript net, and assume gr includes both effects. gr points in the direction of plumbline and is perpendicular to a static water surface – the horizontal plane. Equation (7.15) becomes: r dV 1 rr =-Ñp+gVr -2W´ (7.17a) dt r r r here g is the gnet in (7.17). See figure below. 7-9 Read Symon Sections 7.2 and 7.3, Riegal Pages 144-153! 7.4.2.
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