Up 気象学は「角運動量保存」を誤用する : Hartmann (2016) から 作成: 2022-09-14
更新: 2022-09-14

      Hartmann (2016), §6.3.1
    Wind velocities in the atmosphere are measured in terms of a local Cartesian coordinate system inscribed on a sphere.
    At each latitude (\phi) and longitude λ on a sphere of radius a, the zonal and meridional components of horizontal velocity are defined in the following way (Fig.6.3):

    Figure 6.3
    Local Cartesian coordinates on a sphere and
    the zonal (u), meridional (v), and vertical (w)
    components of the local vector wind velocity.

    \[ u = a\ cos\phi\ \frac{ D \lambda }{ D t } = \rm{ zonal\ or\ eastward\ wind\ speed } \] \[ v = a\ \frac{ D \phi }{ D t } = \rm{ meridional\ or\ northward\ wind\ speed\ \quad \quad } (6.4) \]
    Here D/ Dt represents the material derivative – the temporal tendency that is experienced by an air parcel moving with the flow.

      Hartmann (2016), §6.4
    The general circulation of the atmosphere is heavily constrained by the conservation of angular momentum.
    Angular momentum is the product of mass times the perpendicular distance from the axis of rotation times the rotation velocity.
    The angular momentum about the axis of rotation of Earth can be written as the sum of the angular momentum associated with Earth’s rotation [\(M_Ω\)], plus the angular momentum of zonal air motion measured relative to the surface of Earth [\(M_r\)].
    Because the depth of the atmosphere is so thin compared to the radius of Earth, the altitude in the atmosphere is unimportant for the angular momentum of the air, and we may use a constant radius, \( a \), to describe the distance from the center of Earth.
    The latitude has a strong influence on the angular momentum, however, because increasing latitude decreases the distance to the axis of rotation (Fig.6.15).

    Figure 6.15
    The component of angular momentum about the axis of rotation of Earth.
    The distance to the axis of rotation is the radius of Earth times the cosine of latitude.

    The zonal velocity consists of the velocity of Earth’s surface associated with rotation [\(Ω x (a cos\phi)\)] plus the relative zonal velocity of the wind [\(u\)] .
    \( M_Ω = (Ω\ a\ cos\phi ) \times ( a\ cos\phi ) \)
    \( M_r = u \times ( a\ cos\phi ) \)
    \( M = M_Ω + M_r = ( Ω\ a\ cos\phi + u)\ a\ cos\phi \) ]
    \[ M = ( Ω\ a\ cos\phi + u)\ a\ cos\phi = ( u_{earth} + u)\ a\ cos\phi \quad \quad (6.15) \] The zonal velocity of Earth’s surface at the equator, about \( 465 ms^{−1}\), is very large compared to typical zonal wind speeds in the atmosphere.
    \[ \begin{align} u_{earth} &= Ω\ a\ cos\phi \\ &= 7.292 × 10^{−5}\ rad\ s^{−1} × 6.37 × 10^6\ m ⋅ cos\phi   \\ &= 465\ m\ s^{−1} × cos\phi \\ \end{align} \] The atmospheric angular momentum associated with Earth’s rotation is thus much larger than the angular momentum associated with the zonal winds normally observed in the troposphere.
    When air parcels move poleward in the atmosphere, they retain the same angular momentum unless they exchange angular momentum with other air parcels or with the surface.
    Since the distance to the axis of rotation of Earth decreases as a parcel moves poleward on a level surface, the relative eastward zonal velocity of the parcel must increase to maintain a constant total angular momentum.
    Thus, poleward-moving parcels experience an eastward acceleration relative to Earth’s surface. If a parcel of air moves from one latitude to another while conserving angular momentum, then (6.15) implies a relationship between the zonal velocities the parcel will have at any two latitudes. \[ \begin{align} M &= ( Ω\ a\ cos\phi_1 + u_1)\ a\ cos\phi_1 \\ &= ( Ω\ a\ cos\phi_2 + u_2)\ a\ cos\phi_2 \quad \quad (6.16) \\ \end{align} \]
    If a parcel starts out at the equator with zero relative velocity and moves poleward to another latitude while conserving its angular momentum, we have from (6.16) that
    \[ M = Ω\ a^2 = ( Ω\ a\ cos\phi + u_\phi)\ a\ cos\phi \quad \quad (6.17) \]
    which can be rearranged to yield an expression for the zonal velocity at any other latitude \( u_\phi \),
    \[ u_\phi = Ω\ a\ \frac{ sin^2\phi }{ cos\phi } \quad \quad (6.18) \]
    By substituting numbers into (6.18) we find that a parcel of air with the angular momentum of Earth’s surface at the equator will have a westerly zonal wind speed of \( 134 ms^{−1} \) at 30°N or 30°S.
    This is much greater than the maximum zonally averaged wind speeds in the subtropical jet stream (Fig.6.4), and we infer that the poleward angular momentum transport in the upper, poleward-flowing branch of the Hadley cell is more than adequate to explain the existence of a \(40 ms^{−1} \) jet at 30 ° latitude.
    The interesting part is explaining why the subtropical jet stream is not stronger than it is.

    実際,つぎは間違いである: \[ u = a\ cos\phi\ \frac{ D \lambda }{ D t } = \rm{ zonal\ or\ eastward\ wind\ speed } \]
    球面上の局所デカルト座標系は,この上においた \( u, v, w \) とこれに投影した緯線が,つぎの図のようになる:



    気象学がこんな初歩的なところで間違うとは信じられないことだが,式 \[ u = a\ cos\phi\ \frac{ D \lambda }{ D t } = \rm{ zonal\ or\ eastward\ wind\ speed } \] が揺るがぬ証拠である。

    関連 : 「自転球体上の直進一般にかかる加速度」

  • 引用/参考文献
    • Hartmann, D.L. (2016) : Global Physical Climatology (Sec. Ed.). Elsevier.