Meteorological tsunami generation due to sea-surface pressure change: Three-dimensional theory and synthetics of ocean-bottom pressure change

Although most tsunamis are generated by the sea-bottom deformation caused by earthquakes, some tsunamis are excited by sea-surface pressure changes. This study theoretically investigated tsunamis generated by sea-surface pressure changes and derived the solutions in 3-D space, whereas most past studies employed 2-D equations. Using the solutions, we simulated and visualized the tsunami generation by a growing pressure change. Negative pressure change made the sea surface uplifted inside the source region and negative leading waves were radiated from the source region. We also simulated the tsunami generation when the pressure change at the sea surface moves with almost the same speed as the tsunami propagation velocity. The tsunami height increased with increasing the travel distance including the dispersion eﬀects. The 3-D solutions in this study, including the vertical velocity distribution, indicate that both the tsunami height and the sea-surface pressure changes contribute to the ocean-bottom pressure changes, suggesting the diﬃculty in measuring the tsunami height with ocean-bottom pressure observations. When the pressure change source was characterized by short-wavelength components, the dispersion increased tsunami height more extensively than the non-dispersive tsunamis. The 3-D solutions are necessary for describing the tsunami generation where the long-wave approximations are not applicable in open oceans.


Introduction
While disastrous tsunamis are mostly generated by large earthquakes, some tsunamis are excited by pressure disturbances at sea surfaces caused by meteorological phenomena such as storms and moving convective systems (e.g., Churchill et al. 1995;Monserrat et al. 2006).Such tsunamis are known as meteorological tsunamis or meteotsunamis.The basic generation mechanism of a meteorological tsunami was theoretically investigated in 2-D space with longwave approximations (e.g., Proudman 1929;Greenspan 1956;An et al. 2012;Seo & Liu 2014).These studies predicted that the sea-surface pressure change moving at nearly the same speed as the tsunami speed efficiently amplifies the tsunami height.This amplification is often called the Proudman effect.Meteorological tsunamis have been observed in different regions in the world (e.g., Šepić et al. 2015;Rabinovich et al. 2020).Numerous studies analyzed the records using tsunami simulations reproducing the Proudman effect (e.g., Hibiya & Kajiura 1982).For example, Hibiya & Kajiura (1982) analyzed tide gauge records of meteorological tsunamis observed in Nagasaki bay, Japan.They numerically simulated records by setting a moving pressure change at the sea surface as a force in the long-wave equations.Using detailed distribution of meteorological data such as atmospheric pressure distributions and wind speed distributions, researchers have investigated the sources of meteorological tsunamis (e.g., Williams et al. 2019;Heidarzadeh et al. 2020).
Those analyzed tsunami records were mostly observed near coasts with eyewitness, video captures, and tide gauge records (e.g., Heidarzadeh et al. 2020).The tide gauge records have played an important role for quantitative analyses for the estimation of the tsunami source kinematics or excitation.However, since tide gauges are installed inside bays and harbors, the records are considerably contaminated by strong coastal site effects, which prevents from extracting the details of the generation process.Besides, because the tide gauges are located in very shallow sea and the sampling rate is too low to resolve short-period tsunamis, they hardly capture tsunami dispersion (e.g., Kubota et al. 2020a).Therefore, most past studies used the 2-D non-dispersive tsunami equations for the analyses of the observed data.
Tsunami observations have exhibited impressive growth for past decades due to the development of ocean-bottom pressure gauges, which enabled us to observe tsunamis outside bays and harbors (e.g., Bernard & Meinig 2011;Mungov et al. 2013;Tsushima & Ohta 2014;Rabinovich & Eblé 2015;Kaneda et al. 2015;Kawaguchi et al. 2015;Kanazawa et al. 2016;Aoi et al. 2020).Unlike coastal tide gauges, deep-ocean observations can capture clear tsunami signals (e.g., Grilli et al. 2013;Inazu & Saito 2014;Sandanbata et al. 2018;Kubota et al. 2020b).The high-quality records in open oceans have driven the studies about the dispersion and highresolution tsunami source estimations (e.g., Kirby et al. 2013;Baba et al. 2015;Saito & Kubota 2020).Moreover, the ocean-bottom pressure gauges can catch the tsunamis inside the generation field (e.g., Mikada et al. 2006;Nosov et al. 2007;Mizutani et al. 2020).This drastically changed tsunami research because the generation cannot be described by the 2-D equations that were often used in past tsunami studies.The 3-D fluid theories play a fundamental role for the generation process (Takahashi 1942;Maeda et al. 2013;Lotto & Dunham 2015;Lotto et al. 2017).Saito (2013) investigated the 3-D tsunami generation due to the sea-bottom deformation, providing a theoretical basis for the initial tsunami height distribution for the 2-D tsunami simulations.The study also works as the basis for a unified simulation method for seismic and tsunami waves (Saito & Tsushima 2016;Saito et al. 2019).However, at present, it seems that 3-D studies have not been done for meteorological tsunamis yet.
This study theoretically describes the tsunamis generated by sea-surface pressure changes in an incompressible sea.Particularly, we investigate the ocean-bottom pressure changes during the tsunami generation considering 3-D fluid motions.Besides, the formulation can include the short-wavelength tsunamis exhibiting the dispersion.Section 2 describes a 3-D theory of the tsunami generation.Section 3 visualizes and investigates the tsunami generation due to the seasurface pressure changes based on the solutions.Considering a growing surface pressure change not moving, we simulate the tsunamis to clarify the excitation mechanism.We then simulate the tsunamis generated by a moving surface-pressure change source.We also study the role of dispersion in the meteorological tsunamis.Finally, Section 4 concludes this study.

Fundamental equations
The Cartesian coordinate system (Fig. 1) is used for the formulation, where the  and  axes are in the horizontal plane and the  axis is taken positive downward.The sea surface at rest is located at  = 0. We assume a constant sea depth located at  = ℎ ! .Although many studies described the motion with the spatial description, or Euler description, in fluid dynamics (e.g., Pedlosky 2003), this study employs the material description, or Lagrange description (e.g., Eq. (3.57) in Dahlen & Tromp 1998;Eq. (9.26) in Segall 2010).When neglecting the change of the gravity potential due to the motion, the 3-D linear momentum equation describes the fluid motion, where  is the displacement vector  =  $  7 $ +  %  7 % +  #  7 # ,  is the stress change caused by the motion,  ! is the density,  ! is the gravitational acceleration, and  7 & is the basis vector for the  axis.There is no shear stress in the fluid and the stress tensor is described as where  is the pressure change caused by the motion and  is the identity matrix.We assume an incompressible fluid: At the sea bottom, we set no vertical displacement: At the sea surface, we give the pressure change as The equations of (1) -( 5) describe the fluid motion.Particularly, this study considers the seasurface pressure change Eq. ( 5) as the source.We will find the solutions, (, , , ) and (, , , ).Using  # (, , , ), we derive the tsunami height (, , ) = − # (, , 0, ).

Wavenumber integration method
To find the solutions (, , , ) and (, , , ) that satisfy Eqs. ( 1) -( 5), we use a wavenumber integration method.The displacements and the pressure change are represented as integrals with respect to the wavenumber and the angular frequency: where  = K $ " +  % " .
By substituting Eqs. ( 7), (8), and (9) into the condition of no dilatation (Eq.( 3)), we obtain From the momentum equation (Eq.( 1)) with Eq. ( 10), we obtain the ordinal differential equations with respect to the independent variable : By evaluating the eigenvalues and eignevectors of the 2×2 matrix in Eq. ( 11), we can calculate the propagator matrix () (e.g., Gilbert & Backus 1966;Aki & Richards 2002): This propagator matrix relates the  # and  # at different depths  and  ! as The vertical displacement is zero at the sea bottom (Eq.( 4)).Eq. ( 13) then gives the relation between the values at the surface ( = 0) and the bottom ( = ℎ ! ) as We calculate Eq. ( 14) as Then, we obtain and We can represent  # () and  # () using the propagator matrix as and 2.3 Sudden pressure change at the sea surface We here consider a special case in which a pressure change  !(, ) suddenly occurs at the time  = 0.The surface pressure change  '()*+,-(, , ) is represented as where () is the step function () = 1 when  > +0 and () = 0 when  < −0.A comparison of Eq. (20) with Eq. ( 6) gives the boundary condition at the surface  # H $ ,  % , 0, I as With the propagator matrix (Eq.( 12)) and the boundary condition (Eq.( 21)), Eq. ( 18) is calculated as Eqs. ( 22) and ( 10) give The  # H $ ,  % , , I is then given by Particularly, for the sea bottom ( = ℎ ! ), we obtain 2.4 Integration with respect to the angular frequency With Eq. ( 9), the vertical displacement is represented as The integration with respect to the angular frequency  is conducted with the residue theorem (Appendix A).Then, using the dispersion relation we obtain from Eq. ( 26) .
Similarly, we obtain and Particularly, at the sea bottom, the pressure change is given by .

Impulse responses
Using the solutions ( 28) -( 33), we then derive the solution with respect to the impulsive pressure change at the sea surface given by (, , 0, ) = (, )(). (34) By differentiating the solutions ( 28) -( 33) with respect to time and replacing ̂!H $ ,  % I as ̂!H $ ,  % I = 1, we calculate the wavefield solutions in the case when the surface pressure source is given by Eq. (34).For example, the vertical displacement excited by the source of Eq. ( 34) is calculated from Eq. ( 28) as .
Here we attach a bar on the left-hand side variables as  7 # rather than  # because this impulse response has a dimension that is different from  # .
By differentiating Eq. ( 35) with respect to time, we obtain the vertical velocity as .
Similarly, we obtain the horizontal velocity components: (37) and .
The pressure change in the fluid is In particular, for the sea bottom, (40) Eqs. ( 35) -( 40) are the most important and fundamental solutions in our formulation, because we can obtain any responses with respect to any surface pressure change  '()*+,-H $ ,  % , I with convolution operations.For example, the vertical displacement response is given for an arbitrary sea-surface pressure change source  '()*+,-H $ ,  % , I as 3 Visualization and Interpretations

Surface pressure change at a fixed location
As the simplest case, we consider the surface pressure change not to move.The pressure change starts at  = 0 and increases linearly with duration .The sea-surface pressure source is represented as where and The 2-D Fourier transform of Eq. ( 43) is given by where we set The inverse Fourier transform with Eq. ( 45) gives Using the impulse responses of Eqs. ( 35) -( 40) and the pressure change as source ( 42) and ( 47) in the convolution (see, for example, Eq. ( 41)), we obtain the sea-surface height change and the velocity fields and the ocean-bottom pressure change manuscript submitted to replace this text with name of AGU journal Using these solutions with the depth ℎ != 4 km and setting  != −1 mH2O (~9.8 kPa),  = 20 km, and  = 30 s as a source (Eq.( 42)), we numerically calculate and visualize the tsunami generation and propagation for negative sea-surface pressure change in Fig. 2. given by  = 20 km in Eqs. ( 42) and ( 43).Second panels show tsunami height or vertical upward displacement at the sea surface.Third panels show velocity distribution K $ " +  # " in the sea layer with the depth ℎ != 4 km.Bottom panels show the pressure change at the sea bottom.

Figs 2a and 2b
show that the sea-surface pressure, which we set as a source, decreases with the time until  =  (= 30 s).During this period, the sea-surface height and the velocity in the sea layer change slightly and do not show significant variations (the second and third panels in Figs 2a and 2b).As the time elapses (60 -120s, Figs 2c-2d), the sea-surface height gradually increases.At the time of 250 s (Fig. 2e), the sea-surface uplift reached 1.0 m.While the seasurface height is uplifted inside the pressure source, the surface height is subsided outside the pressure source (x <~-30 km and x >~30 km in the second panels in Figs 2d and 2e) so as to satisfy the water volume conservation.The water volume is displaced horizontally.The displacement propagates as tsunamis.
We should note that the source duration of = 30 s we set in this simulation is shorter than a characteristic time given by 2/ =200 s.This characteristic time is a time scale for a tsunami propagating the distance of the source size or the time for the initial tsunami height to collapse, roughly corresponding to the time the sea-surface height inside the source reaches equilibrium (Fig. 2e).If we set the duration  longer than the characteristic time 2/, the amplitude of the leading tsunamis become smaller (e.g., Saito & Furumura 2009).
The bottom pressure change (the bottom panels in Fig. 2) increases with the time in Figs 2c -2e.The increasing sea-surface height makes the pressure change at the sea bottom less.As more time elapses (Fig. 2f), the sea surface remains uplifted (top second panel in Fig. 2f) to compensate the surface-pressure source, and the horizontally displaced water volume propagates as a tsunami (x <~-30 km and x >~30 km in top second panel in Fig. 2f).The propagating tsunamis are detectable as ocean-bottom pressure changes.

A comparison with the tsunamis generated by an earthquake
The surface pressure change (, , 0, ) =  !(, )() generates tsunamis that are identical to the tsunamis generated by the initial height distribution (, ,  = 0) given by In other words, here, (, , 0) is the equivalent initial tsunami height for the pressure change source  !(, )().The 3-D formulation in this study derives this equivalence between the seasurface pressure change source and the initial height distribution source only for the tsunamis outside the source region.From Eq. ( 33), we represent the tsunamis caused by the surface pressure change as two terms: (53) Note that the first term contributes to only inside the surface pressure change but does not contribute to outside the surface pressure change (i.e., the area where  !(, ) = 0) since it does not propagate.On the other hand, the second term contributes to the tsunamis outside the pressure change since it does propagate as waves due to the factor of cos( !).As a result, the tsunami outside the source region is given by (54) For the tsunamis caused by the sea-bottom deformation (see Appendix B), the tsunami height  C<11<D (, , ) is given by where  • !H $ ,  % I and !H $ ,  % I are the 2-D spatial Fourier transform of the sea-bottom vertical displacement  !(, ) and the corresponding initial tsunami height distribution  !(, ), respectively.Using Eqs. ( 54) and (55), we confirmed that  <=1>&?@>=6A;B@ (, , ) =  C<11<D (, , ) if we set However, we should note that this situation differs for the wavefields inside the source regions.For example, the pressure change at the sea bottom caused by the sea-bottom vertical displacement distribution  !(, ) is given by Eq. (B14) in Appendix B as On the other hand, the pressure change by the corresponding surface-pressure change source is given by Eqs. ( 32) and (56) as (58) We find that that the two pressure changes do not agree, (, , ℎ !, ) ≠  C<11<D (, , ℎ !, ) inside the pressure change source.Such disagreement is also found in the tsunami height and velocity fields inside the pressure change source.These disagreements were not detectable for the past observations in which tsunami sensors were located only outside the source region.However, this difference should appear in recent tsunami observations which are intended to detect tsunamis inside the source region.

Moving surface pressure change
We suppose a moving pressure change with the propagation velocity  ! in the x direction given by The vertical upward displacement, the velocities inside the sea layer, and the pressure change at the sea bottom are obtained by the convolution of the pressure change ( 59) with the impulse responses of Eqs. ( 35) -( 40).The vertical upward displacement is where ̂8( $ ) is the 1-D spatial Fourier transform of  !().Eq. ( 60) corresponds to Eq. (2.4) in Proudman (1929) but is not the same because we give the 3-D case including the dispersion, while 2-D without the dispersion was derived in Proudman (1929).Using  = | $ |, the horizontal and the vertical velocities are given by and The pressure change is given by .
and manuscript submitted to replace this text with name of AGU journal . (64)

Long-wavelength tsuanmis
Using Eqs. ( 60) -( 64), we visualize the wavefields caused by a moving negative pressure change in Fig. 3.The pressure change source is characterized by  != −1 mH2O (=~− 9.8 kPa) and  = 20km in Eq. ( 59).The pressure change source appears at  = 0 s and moves in the positive  direction with the velocity  != 0.18 km/s (= 0.9K !ℎ).At early times (Figs 3a and 3b), the tsunami height is small, less than 0.1 m.As the time increases, the tsunami height increases.At the time 500 s, the maximum height is about 2.5 m, which is considerably larger than the maximum height of 1 m that was observed in the case of a fixed source.This tsunami amplification is due to a moving source, which is often referred to as the Proudman effect (Proudman 1929).59) where the peak pressure change is  != −1 mH2O (~− 9.8 kPa), the duration is  = 30 s, the spatial scale is given by  = 20 km, and the velocity is given by  != 0.18 km/s (= 0.9K !ℎ ! and ℎ != 4 km).Second panels show tsunami height or vertical upward displacement at the sea surface.Third panels show velocity distribution K $ " +  # " in the sea layer with the depth ℎ != 4 km.Bottom panels show the pressure change at the sea bottom.
We further calculate the sea-surface height of Eq.( 60) using approximations to obtain interpretations of Fig. 3. Since  8 () is real, 8(− $ ) = ̂8 * ( $ ) where the asterisk denotes complex conjugation.We calculate Eq. ( 60): (, , ) where Re [⋯ ] indicates the real part of the complex value.We here assume that the pressure change propagation velocity  ! is almost the same as the tsunami phase velocity , or  !~, where we set the phase velocity as  =  | $ | ⁄ =   ⁄ .Setting  =  !+ Δ and Δ ≪  !, we calculate Eq. ( 65): In particular, when the wavelength is much longer than the sea depth, ℎ !≪ 1, the phase velocity  is independent of the wavenumber.Also, using the relation we obtain The first and the second terms indicate the waves propagating along the positive x direction.In particular, the first term increases with increasing the time or increases with the travel distance .As a result, the first term becomes dominant over the second term for long travel distance.
The waveform is given by the spatial derivative of the pressure change distribution.This feature is confirmed in Fig. 3.
Similarly, the pressure change at the sea bottom (Eq.( 64)) is calculated as For the long-wavelength tsunamis ℎ !≪ 1, we obtain (70) The pressure change is given by the sum of two contributions: sea-surface height change  ! !(, , ) and pressure change loaded at the sea surface  '()*+,-(, , ).This indicates that it is not straightforward to measure the tsunami height (, , ) correctly from the ocean-bottom pressure change (, , ℎ !, ).We need to know the sea-surface pressure change  '()*+,-(, , ) to correctly estimate (, , ).For example, for the case of the time 60 s (Fig. 3c), the oceanbottom pressure change is negative −0.5 mH2O.However, the actual tsunami height is positive (0.5 m).

Short-wavelength tsunamis
We next simulate the case of a shorter wavelength ( = 10 km in Eq. ( 59)) pressure change source, where the wavelength of the pressure change is not much longer than the sea depth.Fig. 4 shows the wavefields for various elapsed times.In Fig. 5, in order to compare dispersive and non-dispersive tsunamis, we also plot the wavefields without dispersion by setting  != K !ℎ ! in Eqs. ( 60)-( 64) instead of setting the dispersion relation of Eq. ( 26).Comparing Fig. 4 with Fig. 5, we recognized that the maximum tsunami increased more with increasing the time or the travel distance due to the wave dispersion.Fig. 6 shows the maximum sea-surface height for different travel distances.The maximum height gets much higher than the case without the wave dispersion.It is interesting that the dispersion can enlarge the short-wavelength tsunami height for this moving pressure change source, although the dispersion often reduces the maximum sea-surface height of tsunamis arising from, for example, thrust fault earthquakes (e.g., Saito et al. 2014).A dispersive-tsunami amplification was also found in landslide tsunamis dominated by short-wavelength waves (Baba et al. 2019).59) where the peak pressure change is  != −1 mH2O (~− 9.8 kPa), the duration is  = 30 s, the spatial scale is given by  = 10 km, and the velocity is given by  != 0.18 km/s (= 0.9K !ℎ ! and ℎ != 4 km).Second panels show tsunami height or vertical upward displacement at the sea surface.Third panels show velocity distribution K $ " +  # " in the sea layer with the depth ℎ != 4 km.Bottom panels show the pressure change at the sea bottom.

Conclusions
We formulated 3-D tsunami generation by pressure changes on the sea surface in an incompressible sea.Using 3-D solutions, we can describe the depth-dependent generation field.As a simple case, we simulated the tsunami generation by a growing pressure change at a fixed location.We found that a negative sea-surface pressure change made the sea-surface height uplifted inside the source region and negative leading waves radiated from the source.We also simulated a case when the tsunamis are generated by a moving pressure change.When the pressure change migrates near the tsunami-propagation velocity, tsunami height increases with increasing travel distance.This was originally predicted by Proudman (1929) in the 2-D longwave theory.The solutions in this study are in the 3-D space that can include the depthdependent particle velocities and pressure distributions in the sea.The derived solutions of the pressure changes suggest that both the tsunami height and the sea-surface pressure changes contribute to the ocean-bottom pressure change.This means that we can detect tsunamis by ocean-bottom pressure gauges but need to know the sea-surface pressure change to correctly measure tsunami height by the pressure gauges.Moreover, when the pressure change source was characterized by short-wavelength components, tsunamis showed the dispersion.The dispersion possibly increases tsunami height more extensively than the predictions assuming non-dispersive tsunamis.The solutions derived in this study are useful for recent offshore tsunami observations where the pressure gauges possibly detect the tsunami generation process with meteorological origins.

Figure 1 .
Figure 1.Coordinates used in the formulation.

Figure 2 .
Figure 2. Tsunami generation due to a non-moving surface pressure change for various elapsed times: (a)  = 4, (b) 30, (c) 60, (d) 120, (e) 250, and (f) 500s.Top panels show the pressure change at the surface where the peak pressure change is  != −1 mH2O (~− 9.8 kPa), the duration is  = 30 s and the spatial scale is

Figure 3 .
Figure 3. Tsunami generation due to a moving surface pressure change for various elapsed times: (a)  = 4, (b) 30, (c) 60, (d) 120, (e) 250, and (f) 500s.Top panel shows the pressure change at the surface.The source is given by Eq. (59) where the peak pressure change is  != −1 mH2O (~− 9.8 kPa), the duration is  = 30 s, the spatial scale is given by  = 20 km, and the velocity is given by  != 0.18 km/s (= 0.9K !ℎ ! and ℎ != 4 km).Second panels show tsunami height or vertical upward displacement at the sea surface.Third panels show velocity

Figure 6
Figure 6 Maximum sea-surface height distributions for various moving source cases.The dotted line represents the case when the pressure change source is characterized by  = 20 km in Fig. 3.The solid line represents the case when the pressure change source is characterized by  = 10 km in Fig. 4. The dashed line represents the case when the pressure change source is characterized by  = 10 km but  != K !ℎ ! to exclude the dispersion in Fig. 5.