Atmospheric Dynamics: Forces and Flow Balances

Posted on Dec 14, 2025 in Physics

(1) Fundamental Forces in Rotating Frames

Centrifugal Force

Outward force in a rotating reference frame.

Coriolis Force

Deflection due to Earth’s rotation.

Centripetal Force

Inward force keeping circular motion.

Geostrophic Flow

Balance between Coriolis force and pressure gradient force (PGF).

Inertial Flow

Only Coriolis force acts.

Cyclostrophic Flow

Pressure balances centrifugal force; no Coriolis effect.

Gradient Flow

Curved flow involving all forces.

(2) Centrifugal Effects and Gravity

  • Moment Arm: Longer moment arm (larger R) results in a larger centrifugal acceleration.
  • Rotation Rate: Higher rotation rate ($\Omega\uparrow$) increases centrifugal force.
  • Magnitude: Centrifugal acceleration ($\approx 0.034 \text{ m/s}^2$) is much smaller than gravity ($\approx 9.8 \text{ m/s}^2$).
  • Effective Gravity: Effective gravity equals gravitation plus centrifugal acceleration.
  • Latitude Dependence: Effective gravity depends on position (i.e., latitude).
  • Poles: Centrifugal acceleration is minimum at the poles.
  • Earth Shape: The Earth bulges at the equator due to centrifugal force, establishing its equilibrium shape.

(3) Coriolis Effect on Wind Components

  • Magnitude Dependence: Magnitude is proportional to $\sin(\phi)$, thus depending on latitude.
  • NH ($v’ > 0$): Eastward acceleration.
  • NH ($v’ < 0$): $D’u’/Dt < 0$, resulting in westward acceleration.
  • SH ($v’ > 0$): $D’u’/Dt < 0$, resulting in eastward acceleration.
  • SH ($v’ < 0$): $D’u’/Dt > 0$, resulting in eastward acceleration.
  • NH ($w’ > 0$): $D’u’/Dt < 0$, resulting in westward acceleration.
  • NH ($w’ < 0$): $D’u’/Dt > 0$, resulting in eastward acceleration.
  • Centrifugal Force ($u’ > 0$): Increases centrifugal force, enhancing $C_e$.
  • Centrifugal Force ($u’ < 0$): Reduces $C_e$, leading to an anomalous inward force.
  • NH ($u’ > 0$): Southward acceleration (Coriolis effect).
  • SH ($u’ > 0$): Northward acceleration.
  • Vertical Acceleration: $u’ > 0$ leads to $D’w’/Dt > 0$, causing upward acceleration.

(4) Coriolis Deflection Characteristics

  • NH Deflection: Deflects to the right in the Northern Hemisphere.
  • Effect: Changes wind direction, not wind speed.
  • Latitude: The Coriolis force is smaller at higher latitudes (False in original text, corrected to reflect standard physics: Coriolis force is proportional to $f = 2\Omega\sin\phi$, so it is larger at higher latitudes, except at the equator). Assuming the original intent was to state the force is larger where $f$ is larger: The force is larger at higher latitudes.
  • Wind Speed: More wind speed results in a stronger Coriolis force.

(5) Terms in Equations

  • Third Order Terms: C, D, G.
  • Second Order Terms: A, E.
  • First Order Terms: B, F.
  • Equation Type (First Order Only): Diagnostic.
  • Coriolis Balance PGF: Yes.

(6) Geostrophic Wind Balance

In geostrophic flow:

  • PGF Direction: Points toward low pressure (center).
  • Coriolis Direction: Outward (to the right of motion in NH). It opposes the PGF.
  • Flow Pattern: Wind flows clockwise tangent to isobars (in NH).
  • NH Pressure: Higher pressure is on the right side of the wind.
  • Wind Speed Relation 1: Wind speed is proportional to the pressure gradient magnitude.
  • Wind Speed Relation 2: Wind speed is proportional to $1 / f$ (Coriolis parameter).
  • Latitude Effect: Geostrophic wind speed is smaller at higher latitudes (since $f$ is larger, requiring a smaller gradient to balance). (Corrected based on $V_g = -\frac{1}{f}\frac{\partial\Phi}{\partial n}$ where larger $f$ means smaller $V_g$ for the same gradient.)
  • SH Circulation: Circulation is reversed (counterclockwise) in the Southern Hemisphere.

(7) Prognostic Equations and Balance

  • Equation Type: This equation is prognostic (predicts time evolution), not diagnostic.
  • Small Rossby Number: If the Rossby number is very small, geostrophic balance holds (Coriolis $\approx$ PGF).
  • Acceleration in Geostrophic Flow: No acceleration occurs if $v = v_g$ exactly.

(8) Hydrostatic Approximation Details

  • Order: 1st Order $\rightarrow$ 4th Order (This phrasing is unclear; assuming it refers to the order of the approximation or the resulting equation set).
  • Inherent Error: $0.01\%$ error (!).
  • Dependent Variables: $u, v, \omega, \Phi (Z), \rho, T$.
  • Independent Variables: $x, y, p, t$.
  • Vertical Velocity: $\omega = Dp/Dt$.
  • Total Derivative: $D/Dt = \partial/\partial t + u \partial/\partial x + v \partial/\partial y + \omega \partial/\partial p$.

(9) Flow Regimes and Variables

System Parameters

  • Number of Unknowns: 5.
  • Number of Equations: 5.
  • Unknown Variables: $u, v, \omega, \Phi, T$.
  • Horizontal Wind Speed ($V$): Yes, $V$ is horizontal wind speed.
  • Radius of Curvature ($R$): Yes, $R$ is the radius of curvature.

Flow Types

  • Gradient Flow: Curved large-scale motion.
  • Inertial Flow: Rarely observed.
  • Geostrophic Flow: East-west jet stream.
  • Cyclostrophic Flow: Tornado, dust devil.

Hemisphere Relations (PGF vs. Forces)

  • SH: Coriolis force acts to the left of the PGF.
  • NH: Coriolis force acts to the right of the PGF.

Sign Conventions ($f$ and $R$)

  • SH ($V > 0$): $V$ is positive (clockwise motion). $f$ is negative (Coriolis to the left). $R$ must be negative to satisfy $R = -V/f$.
  • NH ($V > 0$): $V$ is positive (counterclockwise motion). $f$ is positive (Coriolis to the right). $R$ must be positive to satisfy $R = -V/f$.

Case Analysis (Curved Flow Balance)

Case 2 ($R < 0$; $\partial\Phi/\partial n > 0$)

  • Circulation Sense: Clockwise.
  • PGF Direction: Points away from low pressure (outward).
  • PGF Relative to Wind: Points rightward.
  • Centrifugal Force: Points leftward (inward pull balance).

Case 1 ($R > 0$; $\partial\Phi/\partial n < 0$)

  • Circulation Sense: Counterclockwise.
  • PGF Direction: Points toward low pressure (inward).
  • PGF Relative to Wind: Points leftward.
  • Centrifugal Force: Points rightward (outward from center).

(10) Friction and Wind Balance

  • Friction Balance: Yes, a force balance is possible with an adjusted wind vector.
  • Friction/Gradient Balance: Yes, force balance is possible in this case.
  • Friction Effect: Turbulent friction reduces wind speed compared to geostrophic wind, especially near the surface.
  • Friction Redirection: No. Balanced flow redirects wind with a component toward low pressure, not high pressure. (Example: PGF pushes northward; Coriolis opposes it southward.)

(11) Vorticity Equation Terms (Horizontal Advection)

Terms contributing to the time evolution of absolute vorticity ($\zeta + f$):

  • Horizontal Advection: $- \mathbf{v} \cdot \nabla_p (\zeta + f)$.
  • Horizontal Convergence/Divergence (Spin-up of $\zeta$): $- (\zeta + f)(\nabla_p \cdot \mathbf{v})$.
  • Vertical Advection of Relative Vorticity: $- \omega \partial\zeta/\partial p$.
  • Tilting/Twisting: $\hat{k} \cdot (\partial\mathbf{v}/\partial p \times \nabla_p \omega)$.

(12) System Variables and Closure

  • Independent Variables: $x, y, p, t$.
  • Dependent Variables: $u_a, v_a, u_g, v_g, \omega, \Phi, T$.
  • Number of Equations: 7.
  • System Closure: Yes.